http://cantorsattic.info/api.php?action=feedcontributions&user=BartekChom&feedformat=atomCantor's Attic - User contributions [en]2022-08-12T14:44:37ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Worldly&diff=4217Worldly2022-07-02T12:34:40Z<p>BartekChom: /* Otherworldly cardinals */ up to and to $\delta$</p>
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<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of $\text{ZF}$. It follows that $\kappa$ is a [[strong limit]], a [[beth fixed point]] and a fixed point of the enumeration of these, and more.<br />
<br />
* Every [[inaccessible]] cardinal is worldly. (See [[Grothendieck universe]])<br />
* Nevertheless, the least worldly cardinal is [[singular]] and hence not [[inaccessible]]. <br />
* The least worldly cardinal has [[cofinality]] $\omega$.<br />
* Indeed, the next worldly cardinal above any ordinal, if any exist, has [[cofinality]] $\omega$. <br />
* Any worldly cardinal $\kappa$ of uncountable cofinality is a limit of $\kappa$ many worldly cardinals.<br />
<br />
==Degrees of worldliness==<br />
<br />
A cardinal $\kappa$ is ''$1$-worldly'' if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is ''$\alpha$-worldly'' if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$. The cardinal $\kappa$ is ''hyper-worldly'' if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [[inaccessible#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.<!--Also see https://kamerynblog.wordpress.com/2017/01/13/just-how-big-is-the-smallest-inaccessible-cardinal-anyway/--><br />
<br />
The consistency strength of a $1$-worldly cardinal is stronger than that of a worldly cardinal, the consistency strength of a $2$-worldly cardinal is stronger than that of a $1$-worldly cardinal, etc.<br />
<br />
The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.<br />
<br />
==Replacement characterization==<br />
<br />
As long as $\kappa$ is an uncountable cardinal, $V_\kappa$ already satisfies $\text{ZF}\setminus\textrm{Replacement}$ ($\text{ZF}$ without the axiom schema of replacement). So, $\kappa$ is worldly if and only if $\kappa$ is uncountable and $V_\kappa$ satisfies the axiom schema of replacement. More analytically, $\kappa$ is worldly if and only if $\kappa$ is uncountable and for any function $f:A\rightarrow V_\kappa$ definable from parameters in $V_\kappa$ for some $A\in V_\kappa$, $f^{\prime\prime}A\in V_\kappa$ also.<br />
<br />
==Otherworldly cardinals==<br />
<br />
J. D. Hamkins has named a large cardinal property called the ''otherworldly cardinals'': $\kappa$ is otherworldly (to $\lambda$&#x29; if there exists some $\lambda>\kappa$ such that $V_\kappa\prec V_\lambda$. [http://jdh.hamkins.org/otherwordly-cardinals/ "Otherworldly cardinals" (2020)]<br />
<br />
$\kappa$ is called otherworldly up to $\lambda$ if the set of $\mu$ such that $\kappa$ is otherworldly to $\mu$ is cofinal in $\lambda$.<br />
<br />
Otherworldly $\kappa$ have some properties:<br />
* Every otherworldly cardinal is worldly (which played a part in inspiring the name), and also happens to be a limit of worldly cardinals.<br />
* Every otherworldly $\kappa$ is a limit of cardinals $\lambda<\kappa$ such that $Th(V_\lambda&#x29;=Th(V_\kappa&#x29;$.<br />
* Every inaccessible $\delta$ is a limit of otherworldly cardinals.<br />
** In fact, inaccessible $\delta$ is the supremum of the class $\{\kappa\in\delta\mid V_\kappa\prec V_\delta\&#x7d;$.<br />
*** $\delta$ is a limit of cardinals otherworldly up to and to $\delta$.<br />
* A cardinal is otherworldly iff if it is fully correct in a worldly cardinal.<br />
<br />
A cardinal $\kappa$ is ''totally otherworldly'' if for all $\lambda>\kappa$ we have $V_\kappa\prec V_\lambda$ ($\kappa$ is otherworldly to arbitrarily large ordinals&#x29;.<br />
* Every totally otherworldly cardinal is $\Sigma_3$-correct. [http://jdh.hamkins.org/otherwordly-cardinals/#comment-11034]<br />
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[[Category:Large cardinal axioms]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Worldly&diff=4216Worldly2022-07-02T12:19:26Z<p>BartekChom: /* Otherworldly cardinals */ fully correct in a worldly cardinal; it is otherworldly to arbitrarily large ordinals</p>
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<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of $\text{ZF}$. It follows that $\kappa$ is a [[strong limit]], a [[beth fixed point]] and a fixed point of the enumeration of these, and more.<br />
<br />
* Every [[inaccessible]] cardinal is worldly. (See [[Grothendieck universe]])<br />
* Nevertheless, the least worldly cardinal is [[singular]] and hence not [[inaccessible]]. <br />
* The least worldly cardinal has [[cofinality]] $\omega$.<br />
* Indeed, the next worldly cardinal above any ordinal, if any exist, has [[cofinality]] $\omega$. <br />
* Any worldly cardinal $\kappa$ of uncountable cofinality is a limit of $\kappa$ many worldly cardinals.<br />
<br />
==Degrees of worldliness==<br />
<br />
A cardinal $\kappa$ is ''$1$-worldly'' if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is ''$\alpha$-worldly'' if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$. The cardinal $\kappa$ is ''hyper-worldly'' if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [[inaccessible#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.<!--Also see https://kamerynblog.wordpress.com/2017/01/13/just-how-big-is-the-smallest-inaccessible-cardinal-anyway/--><br />
<br />
The consistency strength of a $1$-worldly cardinal is stronger than that of a worldly cardinal, the consistency strength of a $2$-worldly cardinal is stronger than that of a $1$-worldly cardinal, etc.<br />
<br />
The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.<br />
<br />
==Replacement characterization==<br />
<br />
As long as $\kappa$ is an uncountable cardinal, $V_\kappa$ already satisfies $\text{ZF}\setminus\textrm{Replacement}$ ($\text{ZF}$ without the axiom schema of replacement). So, $\kappa$ is worldly if and only if $\kappa$ is uncountable and $V_\kappa$ satisfies the axiom schema of replacement. More analytically, $\kappa$ is worldly if and only if $\kappa$ is uncountable and for any function $f:A\rightarrow V_\kappa$ definable from parameters in $V_\kappa$ for some $A\in V_\kappa$, $f^{\prime\prime}A\in V_\kappa$ also.<br />
<br />
==Otherworldly cardinals==<br />
<br />
J. D. Hamkins has named a large cardinal property called the ''otherworldly cardinals'': $\kappa$ is otherworldly (to $\lambda$&#x29; if there exists some $\lambda>\kappa$ such that $V_\kappa\prec V_\lambda$. [http://jdh.hamkins.org/otherwordly-cardinals/ "Otherworldly cardinals" (2020)]<br />
<br />
Otherworldly $\kappa$ have some properties:<br />
* Every otherworldly cardinal is worldly (which played a part in inspiring the name), and also happens to be a limit of worldly cardinals.<br />
* Every otherworldly $\kappa$ is a limit of cardinals $\lambda<\kappa$ such that $Th(V_\lambda&#x29;=Th(V_\kappa&#x29;$.<br />
* Every inaccessible $\delta$ is a limit of otherworldly cardinals. In fact, inaccessible $\delta$ is the supremum of the class $\{\kappa\in\delta\mid V_\kappa\prec V_\delta\&#x7d;$.<br />
* A cardinal is otherworldly iff if it is fully correct in a worldly cardinal.<br />
<br />
A cardinal $\kappa$ is ''totally otherworldly'' if for all $\lambda>\kappa$ we have $V_\kappa\prec V_\lambda$ ($\kappa$ is otherworldly to arbitrarily large ordinals&#x29;.<br />
* Every totally otherworldly cardinal is $\Sigma_3$-correct. [http://jdh.hamkins.org/otherwordly-cardinals/#comment-11034]<br />
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[[Category:Large cardinal axioms]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Talk:Reflecting_ordinal&diff=4186Talk:Reflecting ordinal2022-06-03T18:06:47Z<p>BartekChom: re</p>
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<div>==Reflecting a conjunctand down==<br />
A property that seems important about reflection is that for any $\Pi_n$-definable class $X$, being $\Pi_n$-reflection on Ord and in $X$ implies being $\Pi_n$-reflecting on $X$. For example, if an ordinal is $\Pi_2$-reflecting and a limit of admissibles, it's also $\Pi_2$-reflecting on the class of limits of admissibles. But I don't know how to add this to the article, since although I've seen a proof I've never seen a published proof [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:10, 30 May 2022 (PDT)<br />
: I added my own argumentation about [[ZFC#Uncountable_transitive_models|uncountable transitive models]], so if you know a proof, writing it is better than what I did. [[User:BartekChom|BartekChom]] ([[User talk:BartekChom|talk]]) 11:06, 3 June 2022 (PDT)</div>BartekChomhttp://cantorsattic.info/index.php?title=Supercompact&diff=4177Supercompact2022-05-29T13:52:29Z<p>BartekChom: /* Laver preparation */ in inner models</p>
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<div>{{DISPLAYTITLE: Supercompact cardinal}}<br />
Supercompact cardinals are best motivated as a generalization of [[measurable]] cardinals, particularly the characterization of measurable cardinals in terms of [[elementary embedding|elementary embeddings]] and strong closure properties. The notion of supercompactness and its consequences was initially developed by Solovay and Reinhardt and further elaborated on by Magidor and Gitik, among many others. Assuming the existence of a supercompact is a very strong assumption and the large cardinal strength of supercompact cardinals is seen in a wide (and bewildering) array of set-theoretic contexts, especially the development of strong forcing axioms and establishing regularity properties of sets of reals. The inner model program has yet to reach the level of a supercompact cardinal and this is considered a prominent open problem in the program itself. Curiously, by results of Woodin, should the inner program reach the level of a supercompact, there is a sense in which it will have reached all greater large cardinals, a startling contrast to previous advances in the program. <br />
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==Formal definition and equivalent characterizations==<br />
<br />
Generalizing the [[elementary embedding]] characterization of measurable cardinal, a cardinal $\kappa$ is ''$\theta$-supercompact'' if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$. Under the [[axiom of choice]], one may assume without loss of generality that $j(\kappa)\gt\theta$. $\kappa$ is then said to be ''supercompact'' if it is $\theta$-supercompact for all $\theta$. It is worth noting that, using this formulation, $H_{\theta^+}$ must be contained in the transitive class $M$. <br />
<br />
There is an alternative formulation that is expressible in $\text{ZFC}$ using certain [[ultrafilter]]s with somewhat technical properties: for $\theta\geq\kappa$, $\kappa$ if $\theta$-supercompact if there is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. $\kappa$ is supercompact if for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$.<br />
<br />
One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. Conversely, the ultrapower by a normal fine measure $U$ on $\mathcal{P}_\kappa(\theta)$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$. <br />
<br />
A third characterization was given by Magidor [Mag71] in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the ''image'' of the critical point of this embedding, rather than the critical point itself: $\kappa$ is supercompact if and only if for every $\eta>\kappa$ there is $\alpha<\kappa$ such that there exists a nontrivial elementary embedding $j:V_\alpha\to V_\eta$ such that $j(\mathrm{crit}(j))=\kappa$. ([[Remarkable]] cardinals could be called virtually supercompact, because one of their definitions is an exact analogue of this one (with forcing extension))<cite>GitmanSchindler:VirtualLargeCardinals</cite><br />
<br />
== Properties ==<br />
<br />
If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ [[filter|normal fine measures]] on $\kappa$, also for every $\lambda\geq\kappa$ there are $2^{2^{\lambda^{<\kappa}}}$ normal fine measures on $\mathcal{P}_\kappa&#40;\lambda)$.<br />
<br />
Every supercompact has [[Mitchell order]] $&#40;2^\kappa)^+\geq\kappa^{++}$.<br />
<br />
If $\kappa$ is $\lambda$-supercompact then it is also $\mu$-supercompact for every $\mu<\lambda$. If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha<\kappa$ that is $\gamma$-supercompact for all $\gamma<\kappa$ &#40;if any exists) is also $\lambda$-supercompact. In the same vein, for every cardinals $\kappa<\lambda$, if $\lambda$ is supercompact and $\kappa$ is $\gamma$-supercompact for all $\gamma<\lambda$, then $\kappa$ is also supercompact.<br />
<br />
''Laver's theorem'' asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $|tc&#40;x)|\leq\lambda$ there exists a normal fine measure $U$ on $\mathcal{P}_\kappa&#40;\lambda)$ such that $j_U&#40;f)&#40;\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc&#40;x)$ is the ''transitive closure'' of $x$ &#40;i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.<br />
<br />
Without the [[axiom of choice]] $\omega_1$ can be supercompact. &#40;[https://link.springer.com/article/10.1007/BF02771215 Jech, 1968], Takeuti, 1970, after [https://eprints.illc.uva.nl/id/eprint/689/1/X-2013-02.text.pdf])<br />
<br />
== Supercompact cardinals and forcing ==<br />
<br />
=== The continuum hypothesis and supercompact cardinals ===<br />
<br />
If $\kappa$ is $\lambda$-supercompact and $2^\alpha=\alpha^{+}$ for every $\alpha<\kappa$, then $2^\alpha=\alpha^{+}$ for every $\alpha\leq\lambda$. Consequently, if the [[GCH|generalized continuum hypothesis]] holds below a supercompact cardinal, then it holds everywhere.<br />
<br />
The existence of a supercompact implies the consistency of the failure of the ''singular cardinal hypothesis'', i.e. it is consistent that the generalized continuum hypothesis fails at a strong limit singular cardinal. It also implies the consistency of the failure of the $\text{GCH}$ at a measurable cardinal.<br />
<br />
By combining results of Magidor, Shelah and Gitik, one can show that the existence of a supercompact also implies the existence of a [[forcing|generic extension]] in which $2^{\aleph_\alpha}<\aleph_{\omega_1}$ for all $\alpha<\omega_1$, but also $2^{\aleph_{\omega_1}}>\aleph_{\omega_1+\alpha+1}$ for any prescribed $\alpha<\omega_2$. Similarly, one can have a generic extension in which the $\text{GCH}$ holds below $\aleph_\omega$ but $2^{\aleph_\omega}>\aleph_{\omega+\alpha+1}$ for any prescribed $\alpha<\omega_1$.<br />
<br />
Woodin and Cummings furthermore showed that if there exists a supercompact, then there is a generic extension in which $2^\kappa=\kappa^{++}$ for ''every'' cardinal $\kappa$, i.e. the $\text{GCH}$ fails ''everywhere''(!).<br />
<br />
The [[ultrapower axiom]], if consistent with a supercompact, implies that the $\text{GCH}$ holds above the least supercompact.<br />
<br />
=== Laver preparation ===<br />
<br />
''Indestructibility, including the Laver diamond.'' [Laver 78]<br />
<br />
Some properties usually obtained by forcing are possible in [[inner model]]s, for example<cite>ApterGitmanHamkins2012:InnerModelsUsuallyForcing</cite>:<br />
* (theorem 14) If there is a supercompact cardinal, then there are inner models with an [[indestructible]] supercompact cardinal $κ$ such that<br />
** $2^κ = κ^+$<br />
** $2^κ = κ^{++}$<br />
** Moreover, for every cardinal $θ$, such inner models $W$ can be found for which also $W^θ ⊆ W$.<br />
<br />
=== Proper forcing axiom ===<br />
<br />
Baumgartner proved that if there is a supercompact cardinal, then the [[proper forcing axiom]] holds in a [[forcing]] extension. $\text{PFA}$'s strengthening, $\text{PFA}^{+}$, is also consistent relative to the existence of a supercompact cardinal.<br />
<br />
=== Martin's Maximum ===<br />
<br />
== Relation to other large cardinals ==<br />
<br />
Every cardinal $\kappa$ that is $2^\kappa$-supercompact is a stationary limit of [[superstrong]] cardinals, but need not be superstrong itself. In fact $2^\kappa$-supercompact are stationary limits of quasicompacts, themselves stationary limits of 1-[[extendible|extendibles]].<br />
<br />
If $\theta=\beth_\theta$ then every $\theta$-supercompact cardinal is [[strong|$\theta$-strong]]. This is because $H_{\theta^+}\in M$ so $H_{\theta^+}\subset M$ by transitivity and $V_\theta\subset H_\theta\in M$ so $V_\theta\subset M$, as desired.<br />
<br />
If a cardinal $\theta$-supercompact then it also $\theta$-[[strongly compact]]. Consequently, every supercompact cardinal is also strongly compact. It is consistent with $\text{ZFC}$ that every strongly compact cardinal is also supercompact, but it is not currently known whether the existence of a strongly compact cardinal is equiconsistent with the existence of a supercompact cardinal. The [[ultrapower axiom]] gives a positive answer to this, but itself isn't known to be consistent with the existence of a supercompact in the first place. If $\kappa$ is supercompact, then there is a forcing extension in which $\kappa$ remains supercompact and is also the least strongly compact cardinal. If there exists a measurable cardinal that is a limit of strongly compact cardinals, then the least such cardinal is strongly compact but not supercompact, in fact not even $2^\kappa$-supercompact.<br />
<br />
Under the [[axiom of determinacy]], $\omega_1$ is <$\Theta$-supercompact, where $\Theta$ is at least an [[aleph fixed point]], and under $V=L(\mathbb{R})$ is even weakly hyper-[[Mahlo]]. The existence of a supercompact cardinals also implies the axiom $\text{AD}^{L(\mathbb{R})}$.<br />
<br />
If $\kappa$ is $|V_{\kappa+\eta}|$-supercompact with $\eta<\kappa$ then it is preceeded by a stationary set of $\eta$-[[extendible]] cardinals. If $\kappa$ is $(\eta+2)$-extendible then it is $|V_{\kappa+\eta}|$-supercompact. The least supercompact is not 1-extendible, in fact any cardinal that is both supercompact and 1-extendible is preceeded by a stationary set of cardinals that are both supercompact and limits of supercompact cardinals.<br />
<br />
The least supercompact is larger than the least [[huge]] cardinal (if such a cardinal exists). It is also larger than the least n-huge cardinal, for all n. If $\kappa$ is supercompact and there is an n-huge cardinal above $\kappa$, then there are $\kappa$-many n-huge cardinals below $\kappa$.<br />
<br />
From <cite>Bagaria2012:CnCardinals</cite>:<br />
* If κ is $2^κ$-supercompact and belongs to $C^{(n)}$, then there is a $κ$-complete normal [[ultrafilter]] $U$ over $κ$ such that the set of $C^{(n)}$-[[superstrong]] cardinals smaller than $κ$ belongs to $U$.<br />
* $VP(Π_1) \iff VP(κ, Σ_2)$ for some $κ \iff$ There exists a supercompact cardinal. ($VP$ — [[Vopenka|Vopěnka's principle]])<br />
* $VP(\mathbf{Π_1}) \iff VP(κ, \mathbf{Σ_2})$ for a proper class of cardinals $κ \iff$ There is a proper class of supercompact cardinals.<br />
<br />
== Enhanced supercompact cardinals ==<br />
(from <cite>Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge</cite>)<br />
<br />
A cardinal κ is called ''enhanced supercompact'' iff there is a [[strong]] cardinal $θ > κ$ such that for every cardinal $λ > θ$, there is a $λ$-supercompactness embedding $j : V → M$ such that $θ$ is strong in $M$.<br />
<br />
If the cardinal $δ$ is Woodin for supercompactness (i.e. [[Vopenka|Vopěnka]]), then there are unboundedly many cardinals $κ < δ$ such that $κ$ is a limit of cardinals $η$ such that there exists an inaccessible cardinal $β$ such that $η < β < κ$, and $V_β \models$ “$η$ is enhanced supercompact”.<br />
<br />
The relationship between [[extendible]], [[hypercompact]] and enhanced supercompact cardinals is not known.<br />
<br />
== $C^{(n)}$-supercompact ==<br />
(from <cite>Bagaria2012:CnCardinals</cite>, 2019 extended arXiv version)<br />
<br />
Definitions:<br />
* A cardinal $κ$ is '''[[correct|$θ$-$C^{(n)}$-supercompact]]''' (for $θ > κ$) iff there is an elementary embedding $j : V → M$ with transitive $M$ and such that $crit(j) = κ$, $j(κ) > λ$, $M^\theta\subset M$ and $j(κ) ∈ C^{(n)}$.<br />
* $κ$ is '''$C^{(n)}$-supercompact''' iff it is $θ$-$C^{(n)}$-supercompact for every $θ > κ$.<br />
<br />
Equivalently:<br />
* $κ$ is $θ$-$C^{(n)}$-supercompact iff there are $μ$, $E$, $Y$ and $ζ$ such that<br />
** $μ ∈ C^{(n)}$<br />
** $θ, E, Y ∈ V_μ$<br />
** $Y$ is transitive<br />
** $[Y]^{≤θ} ⊆ Y$<br />
** $V_μ \models$<br />
*** $E$ is an extender over $V_ζ$ with critical point $κ$ and support $Y$<br />
*** $j_E [Y] ⊆ Y$<br />
*** $j_E (κ) > θ$<br />
*** $j_E (κ) ∈ C^{(n)}$<br />
<br />
Properties:<br />
* The notion of $θ$-$C^{(n)}$-supercompactness, unlike $θ$-supercompactness, cannot be formulated in terms of normal measures on $\mathcal{P}_κ (θ)$.<br />
** One must use, e.g., long [[extender]]s (above) to see that, for $n ≥ 1$, “$κ$ is $θ$-$C^{(n)}$-supercompact” is $Σ_{n+1}$ expressible, so<br />
** “κ is $C^{(n)}$-supercompact” is $Π_{n+2}$ expressible, so<br />
** if $κ$ is $C^{(n)}$-supercompact, $α ∈ C^{(n+1)}$ and $α > κ$, then $V_α \models$ “$κ$ is $C^{(n)}$-supercompact”.<br />
* “$∃_κ (κ$ is $C^{(n)}$-supercompact$)$” is $Σ_{n+3}$-expressible, so the first $C^{(n)}$-supercompact cardinal does not belong to $C^{(n+3)}$.<br />
* The first $C^{(n)}$-supercompact cardinal is smaller than the first $C^{(n+1)}$-[[extendible]] cardinal (provided both exist).<br />
* Every $C^{(n)}$-[[huge|superhuge]] cardinal is $C^{(n)}$-supercompact.<br />
* Assuming [[rank into rank|$\mathrm{I3}(κ, δ)$]], if $δ$ is a limit cardinal (instead of a successor of a limit cardinal – Kunen’s Theorem excludes other cases), it is equal to $sup\{j^m(κ) : m ∈ ω\}$ where $j$ is the elementary embedding. Then $κ$ and $j^m(κ)$ are $C^{(n)}$-supercompact (inter alia) in $V_δ$, for all $n$ and $m$.<br />
<br />
Unknowns:<br />
* Does the first $C^{(1)}$-supercompact cardinal belong to $C^{(3)}$?<br />
* Do the $C^{(n)}$-supercompact cardinals form a hierarchy in a strong sense, that is, is the first $C^{(n)}$-supercompact cardinal smaller than the first $C^{(n+1)}$-supercompact cardinal, for all $n$?<br />
* Is every $C^{(n)}$-extendible cardinal $C^{(n)}$-supercompact?<br />
* Is the first $C^{(n)}$-extendible cardinal greater than the first $C^{(n)}$-supercompact cardinal?<br />
<br />
{{stub}}<br />
{{references}}<br />
<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Indestructible&diff=4176Indestructible2022-05-29T13:46:11Z<p>BartekChom: Redirected page to Forcing#Indestructibility</p>
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<div>#REDIRECT [[Forcing#Indestructibility]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Forcing&diff=4175Forcing2022-05-29T13:43:41Z<p>BartekChom: /* Separativity */ === Indestructibility ===</p>
<hr />
<div>'''Forcing''' is a method for extending a transitive model $M$ of $\text{ZFC}$ (the ''ground [[model]]'') by adjoining a new set $G$ (the ''generic set'') to produce a new, larger model $M[G]$ called a ''generic extension''. In short, the set $G$ can be constructed a certain way using a partially ordered set (poset) $(\mathbb{P},\leq)\in M$ (the ''forcing notion'') so that the following holds:<br />
<br />
* '''(Generic Model Theorem).''' There exists a unique transitive model $M[G]$ of $\text{ZFC}$ that includes $M$ (as a subset) and contains $G$ (as an element), has the same ordinals as $M$, and any transitive model of $\text{ZFC}$ also including $M$ and containing $G$ includes $M[G]$ (i.e. $M[G]$ is minimal).<br />
<br />
The elements of the forcing notion $\mathbb{P}$ will be called the ''conditions''. The order $p\leq q$, for $p,q\in\mathbb{P}$, is to be interpreted as "$p$ is stronger than $q$" or as "$p$ implies $q$". $G$ will be a special subset of $\mathbb{P}$ said to be ''generic over $M$'' and satisfying some requirements. The choice of $\mathbb{P}$ and of $\leq$ will determine what is true of false in $M[G]$. A special relation called the ''forcing relation'' is defined, which links the conditions to the formulas they will force. It is very important to note that this relation can be defined from within the ground model $M$.<br />
<br />
While the usual definition of forcing only works for transitive countable models $M$ of $\text{ZFC}$, it is customary to "take $V$ as the ground model", and pretend there exists a generic $G\subseteq\mathbb{P}$. Every statement about the generic extension $V[G]$ can be thought as a statement in the forcing relation: that relation being definable within the ground model, this method can be thought as working within the ground model $M$, with $V[G]$ being, in some way, $M[G]$ as seen from within the ground model $M$.<br />
<br />
Forcing was first introduced by Paul Cohen as a way of proving the consistency of the failure of the [[GCH|continuum hypothesis]] with $\text{ZFC}$. He also used it to prove the consistency of the failure of the [[axiom of choice]], albeit the proof is more indirect: if $M$ satisfies choice, then so does $M[G]$, so $\neg AC$ cannot be forced directly, though it is possible to extract a submodel of $M[G]$ (for a particular generic extension) in which choice fails.<br />
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In particular, an inner model (a class-sized transitive model (of ZFC or a weaker theory) containing all ordinals) can be a ground of $V$.<br />
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== Definitions and some properties ==<br />
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Let $(\mathbb{P},\leq)$ be a partially ordered set, the ''forcing notion''. Sometimes $\leq$ can just be a preorder (i.e. not necessarily antisymmetric). The elements of $\mathbb{P}$ are called ''conditions''. We will assume $\mathbb{P}$ has a maximal element $1$, i.e. one has $p\leq 1$ for all $p\in\mathbb{P}$. This element isn't necessary if one uses the definition using Boolean algebras presented below, but is useful when trying to construct $M[G]$ without using Boolean algebras.<br />
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Two conditions $p,q\in P$ are ''compatible'' if there exists $r\in\mathbb{P}$ such that $r\leq p$ and $r\leq q$. They are ''incompatible'' otherwise. A set $W\subseteq\mathbb{P}$ is an ''antichain'' if all its elements are pairwise incompatible.<br />
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=== Genericity ===<br />
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A nonempty set $F\subseteq\mathbb{P}$ is a ''[[filter]] on $\mathbb{P}$'' if all of its elements are pairwise compatible and it is closed under implications, i.e. if $p\leq q$ and $p\in F$ then $q\in F$.<br />
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One says that a set $D\subseteq\mathbb{P}$ is ''dense'' if for all $p\in\mathbb{P}$, there is $q\in D$ such that $q\leq p$ (i.e. $q$ ''implies'' $p$). $D$ is ''open dense'' if additionally $q\leq p$ and $p\in D$ implies $q\in D$. $D$ is ''predense'' if every $p\in\mathbb{P}$ is compatible with some $q\in D$.<br />
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Given a collection $\mathcal{D}$ of dense subsets of $\mathbb{P}$, one says that a filter $G$ is '''$\mathcal{D}$-generic''' if it intersects all sets $D\in\mathcal{D}$, i.e. $D\cap G\neq\empty$.<br />
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Given a transitive model $M$ of $\text{ZFC}$ such that $(\mathbb{P},<)\in M$, we say that a filter $G\subseteq\mathbb{P}$ is '''$M$-generic''' (or $\mathbb{P}$-generic in $M$, or just generic) if it is $\mathcal{D}_M$-generic where $\mathcal{D}_M$ is the set of all dense subsets of $\mathbb{P}$ in $M$.<br />
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In the above definitions, ''dense'' can be replaced with ''open dense'', ''predence'' or ''a maximal antichain'', and the resulting notion of genericity would be the same.<br />
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In most cases, if $G$ is $\mathbb{P}$-generic over $M$ then $G\not\in M$. The Generic Model Theorem mentioned above says that there is a minimal model $M[G]\supseteq M$ with $M[G]\models\text{ZFC}$, $G\in M[G]$, and if $M\models$ "$x$ is an ordinal" then so does $M[G]$.<br />
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=== $\mathbb{P}$-names and interpretation by $G$ ===<br />
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Using transfinite recursion, define the following cumulative hierarchy:<br />
* $V^\mathbb{P}_0=\empty$, $V^\mathbb{P}_\lambda = \bigcup_{\alpha<\lambda}V^\mathbb{P}_\alpha$ for limit $\lambda$<br />
* $V^\mathbb{P}_{\alpha+1} = \mathcal{P}(V^\mathbb{P}_\alpha\times\mathbb{P})$<br />
* $V^\mathbb{P} = \bigcup_{\alpha\in\mathrm{Ord}}V^\mathbb{P}_\alpha$<br />
Elements of $V^\mathbb{P}$ are called ''$\mathbb{P}$-names''. Every nonempty $\mathbb{P}$-name is of a set of pairs $(n,p)$ where $n$ is another $\mathbb{P}$-name and $p\in\mathbb{P}$.<br />
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Given a filter $G\subseteq\mathbb{P}$, define the ''interpretation of $\mathbb{P}$-names'' by $G$: Given a $\mathbb{P}$-name $x$, let $x^G=\{y^G : ((\exists p\in G)(y,p)\in x)\}$. Letting $\breve{x}=\{(\breve{y},1):y\in x\}$ for every set $x$ be the ''canonical name'' for $x$, one has $\breve{x}^G=x$ for all $x$.<br />
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Let $M$ be a transitive model of $\text{ZFC}$ such that $(\mathbb{P},\leq)\subseteq M$. Let $M^\mathbb{P}$ be the $V^\mathbb{P}$ constructed in $M$. Given a $M$-generic filter $G\subseteq\mathbb{P}$, we can now define the generic extension $M[G]$ to be $\{x^G : x\in M^\mathbb{P}\}$. This $M[G]$ satisfies the Generic Model Theorem.<br />
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=== The forcing relation ===<br />
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Define the ''forcing language'' to be the first-order language of set theory augmented by a constant symbol for every $\mathbb{P}$-name in $M^\mathbb{P}$. Given a condition $p\in\mathbb{P}$, a formula $\varphi(x_1,...,x_n)$ and $x_1,...,x_n \in M^\mathbb{P}$, we say that '''$p$ forces $\varphi(x_1,...,x_n)$''', denoted $p\Vdash_ {M,\mathbb{P}}\varphi(x_1,...,x_n)$ if for all $M$-generic filter $G$ with $p\in G$ one has $M[G]\models\varphi(x_1^G,...,x_n^G)$. There exists an "internal" definition of $\Vdash$, i.e. a definition formalizable in $M$ itself, by induction on the complexity of the formulas of the forcing language.<br />
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The '''Forcing Theorem''' asserts that if $\sigma$ is a sentence of the forcing language, $M[G]$ satisfies $\sigma$ if and only if some condition $p\in G$ forces $\sigma$.<br />
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The forcing relation has the following properties, for all $p,q\in\mathbb{P}$ and formulas $\varphi,\psi$ of the forcing language:<br />
* $p\Vdash\varphi\land q\leq p\implies q\Vdash\varphi$<br />
* $p\Vdash\varphi\implies\neg(p\Vdash\neg\varphi)$<br />
* $p\Vdash\neg\varphi\iff\neg\exists q\leq p(q\Vdash\varphi)$<br />
* $p\Vdash(\varphi\land\psi)\iff(p\Vdash\varphi\land p\Vdash\psi)$<br />
* $p\Vdash\forall x\varphi\iff\forall x\in M^\mathbb{P}(p\Vdash\varphi(x))$<br />
* $p\Vdash(\varphi\lor\psi)\iff\forall q\leq p\exists r\leq q(r\Vdash\varphi\lor r\Vdash\psi)$<br />
* $p\Vdash\exists x\varphi\iff\forall q\leq p\exists r\leq q\exists x\in M^\mathbb{R}(r\Vdash\varphi(x))$<br />
* $p\Vdash\exists x\varphi\implies\exists x\in M^\mathbb{P}(p\Vdash\varphi(x))$<br />
* $\forall p\exists q\leq p (q\Vdash\varphi\lor q\Vdash\neg\varphi)$<br />
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=== Separativity ===<br />
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A forcing notion $&#40;\mathbb{P},\leq)$ is ''separative'' if for all $p,q\in\mathbb{P}$, if $p\not\leq q$ then there exists a $r\leq p$ incompatible with $q$. Many notions aren't separative, for example if $\leq$ is a linear order than $&#40;\mathbb{P},\leq)$ is separative iff $\mathbb{P}$ has only one element. However, every notion $&#40;\mathbb{P},\leq)$ has a unique &#40;up to isomorphism) ''separative quotient'' $&#40;\mathbb{Q},\preceq)$, i.e. a notion $&#40;\mathbb{Q},\preceq)$ and a function $i:\mathbb{P}\to\mathbb{Q}$ such that $x\leq y\implies i&#40;x)\preceq i&#40;y)$ and $x, y$ are compatible iff $i&#40;x),i&#40;y)$ are compatible. This name comes from the fact that $\mathbb{Q}=&#40;\mathbb{P}/\equiv)$ where $x\equiv y$ iff every $z\in P$ is compatible with $x$ iff it is compatible with $y$. The order $\preceq$ on the equivalence classes is "$[x]\preceq[y]$ iff all $z\leq x$ are compatible with $y$". Also $i&#40;x)=[x]$. It is sometimes convenient to identify a forcing notion with its separative quotient.<br />
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=== Indestructibility ===<br />
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A large cardinal is called ''&#40;Laver) indestructible'' iff it keeps the given property after any $<κ$-directed closed forcing. <cite>ApterGitmanHamkins2012:InnerModelsUsuallyForcing</cite><sup>above Test Question 4</sup><br />
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== Boolean algebras ==<br />
''To be expanded.''<br />
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It is sometimes convenient to take the forcing notion $\mathbb{P}$ to be a Boolean algebra $\mathbb{B}$.<br />
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== Consistency proofs ==<br />
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Let $T_1$ and $T_2$ be some recursively enumerable enumerable extensions of $\text{ZFC}$ (possibly $\text{ZFC}$ itself). The existence of a countable transitive model $M$ of the theory $T_1$ is equivalent to the assertion that $T_1$ is consistent. When we construct a generic extension $M[G]$ satisfying $T_2$ from a countable transitive model $M$ of $T_1$, we prove the consistency of $T_2$ (since we prove it has a set model) only from the consistency of $T_1$, i.e. we prove $\text{Con}(T_1)\implies\text{Con}(T_2)$.<br />
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For instance, by following Cohen's construction of a generic extension statisfying $\text{ZFC+}\neg\text{CH}$ from a model of $\text{ZFC}$, we prove that $\text{Con}(\text{ZFC})\implies\text{Con}(\text{ZFC+}\neg\text{CH})$. It follows that if $\text{ZFC}$ is consistent then it cannot prove $\text{CH}$, as otherwise $\text{ZFC+}\neg\text{CH}$ would be inconsistent, contradicting the above implication proved by forcing.<br />
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Those implications between the consistencies of different theories are the ''relative consistency results'' set theorists are often interested in. The subsection below provides many more examples of consistency results, where the theory $T_1$ above is often $\text{ZFC}$ augmented by large cardinal axioms. <br />
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=== Other examples of consistency results proved using forcing ===<br />
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In the following examples, the generated generic extensions satisfy the axiom of choice unless indicated otherwise.<br />
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* '''Easton's theorem:''' Let $M$ be a transitive set model of $\text{ZFC+GCH}$. Let $F$ be an increasing function in $M$ from the set of $M$'s regular cardinals to the set of $M$'s cardinals, such that for all regular $\kappa$, $\mathrm{cf}F(\kappa)>\kappa$. Then there is a generic extension $M[G]$ of $M$ with the same cardinals and cofinalities such that $M[G]\models\text{ZFC+}\forall\kappa($if $\kappa$ is regular then $2^\kappa=F(\kappa)$).<br />
* '''Violating the Singular Cardinal Hypothesis at $\aleph_\omega$:''' Assume there is a [[measurable]] cardinal of [[Mitchell order]] $o(\kappa)=\kappa^{++}$. Then there is a generic extension in which $\kappa=\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}$. The hypothesis used here is optimal: in term of consistency strength, no less than a measurable of order $\kappa^{++}$ can produce a model where $\text{SCH}$ fails.<br />
* '''Violating the Singular Cardinal Hypothesis everywhere:''' It is consistent relative to the existence of a $(\delta+2)-$[[strong]] cardinal $\delta$ that $2^\kappa=\kappa^+$ for every successor $\kappa$ but $2^\kappa=\kappa^{++}$ for every limit cardinal $\kappa$.<br />
* '''Violating the Generalized Continuum Hypothesis everywhere:''' It is consistent relative to the existence of a $(\delta+2)-$strong cardinal $\delta$ that $2^\kappa=\kappa^{++}$ for every $\kappa$, i.e. $\text{GCH}$ fails everywhere.<br />
* '''Large cardinal properties of $\aleph_1$:''' Let $\kappa$ be measurable/[[supercompact]]/[[huge]]. Then there is a (sub)model (of a generic extension) satisfying $\text{ZF(+}\neg\text{AC)}$ in which $\kappa=\aleph_1$ and $\omega_1$ is measurable/supercompact/huge (by the ultrafilter characterizations, not by the elementary embedding characterizations.)<br />
* '''Singularity of every uncountable cardinal:''' It is consistent relative to the existence of a proper class of [[strongly compact]] cardinals there is model of $\text{ZF}$ in which (the axiom of choice does not hold and) every uncountable cardinal is singular and has cofinality $\omega$. The existence of a such model also implies that the [[axiom of determinacy]] holds in the $L(\mathbb{R})$ of some forcing extension of $\text{HOD}$.<br />
* '''[[Projective|Regularity properties]] of all sets of reals:''' Assume there is an [[inaccessible]] cardinal $\kappa$. Then there is a (sub)model (of a generic extension) that satisfies $\text{ZF+DC+}\neg\text{AC}$ and in which $\kappa=2^{\aleph_0}$ and every set of reals is Lebesgue measurable, has the Baire property and the perfect subset property. There is also a generic extension in which choice holds and every [[projective]] set of reals has those properties.<br />
* '''Real-valued measurability of the continuum:''' Assume there is a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$ and $2^{\aleph_0}$ is real-valued measurable (and thus weakly inaccessible, weakly hyper-[[Mahlo]], etc.)<br />
* '''Precipitousness of the [[filter|nonstationary ideal]] on $\omega_1$:''' Assume there is a measurable cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_1$ and the nonstationary ideal on $\omega_1$ is precipitous.<br />
* '''Saturation of the nonstationary ideal on $\omega_1$:''' Assume there is a [[Woodin]] cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_2$ the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated.<br />
* '''Saturation of an ideal on the continuum:''' Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$, there is a $2^{\aleph_0}$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ and there isn't any $\lambda$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ for every infinite $\lambda<2^{\aleph_0}$.<br />
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Some other applications of forcing:<br />
<!--* There is a generic extension in which there is a cardinal $\kappa$ such that $2^{\mathrm{cf}(\kappa)}<\kappa<\kappa^+<\kappa^{\mathrm{cf}(\kappa)}<2^\kappa$.--><br />
* It is consistent relative to the existence of an inaccessible cardinal that there are no Kurepa trees.<br />
* Let $\kappa$ be a [[superstrong]] cardinal. Let $V[G]$ be the generic extension of $V$ by the Lévy collapse $\mathrm{Coll}(\aleph_0,<\kappa)$. Then there is a nontrivial [[elementary embedding]] $j:L(\mathbb{R})\to(L(\mathbb{R}))^{V[G]}$.<br />
* Let $\kappa$ be a superstrong cardinal. There exists a $\omega$-distributive $\kappa$-c.c. notion of forcing $(\mathbb{P},\leq)$ such that in $V^\mathbb{P}$, $\kappa=\aleph_2$ and there exists a normal $\omega_2$-saturated $\sigma$-complete ideal on $\omega_1$.<br />
* Let $\kappa$ be a [[weakly compact]] cardinal. Then there is a generic extension in which $\kappa=\aleph_2$ and $\omega_2$ has the tree property. In fact, if there is infinitely many weakly compact cardinals then in a generic extension $\omega_{2n}$ has the tree property for every $n$. [http://logika.ff.cuni.cz/radek/papers/Friedman_Honzik_treeprop_revised.pdf]<br />
* It is consistent relative to the existence of infinitely many supercompact cardinals that there exists infinitely many cardinals $\delta$ above $2^{\aleph_0}$ such that both $\delta$ and $\delta^+$ have the tree property. Also, the [[axiom of projective determinacy]] holds in any such model.<br />
* Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa$ remains weakly compact, there is a $\kappa^+$-saturated $\kappa$-complete ideal on $\kappa$ but there isn't any $\kappa$-saturated $\kappa$-complete ideal on $\kappa$. One can replace "$\kappa$ is weakly compact" by "$\kappa$ is weakly inaccessible and $\kappa=2^{\aleph_0}$".<br />
* It is consistent relative to a supercompact cardinal that there is an inaccessible cardinal $\kappa$, a cardinal $\lambda>\kappa$ and a stationary set $S\subseteq\mathcal{P}_\kappa(\lambda)$ that cannot be partitioned into $\kappa^+$ disjoint stationary subsets.<br />
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== Types of forcing ==<br />
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=== Chain conditions, distributivity, closure and property (K) ===<br />
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A forcing notion $(\mathbb{P},\leq)$ satisfies the ''$\kappa$-chain condition'' ($\kappa$-c.c.) if every antichain of elements of $\mathbb{P}$ has cardinality less than $\kappa$. The $\omega_1$-c.c. is called the ''countable chain condition'' (c.c.c.). An important feature of chain conditions is that if $(\mathbb{P},\leq)$ satisfies the $\kappa$-c.c. then if $\kappa$ is regular in $M$ then it will be regular in $M[G]$. Since the $\kappa$-c.c. implies the $\lambda$-c.c. for all $\lambda\geq\kappa$, it follows that the $\kappa$-c.c. implies all regular cardinals $\geq|\mathbb{P}|^+$ will be preserved, and in particular the c.c.c. implies all cardinals and cofinalities of $M$ will be preserved in $M[G]$ for all $M$-generic $G\subseteq\mathbb{P}$.<br />
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Let $\kappa$ be a regular uncountable cardinal. If $(\mathbb{P},\leq)$ is a $\kappa$-c.c. notion of forcing then every club $C\in M[G]$ of $\kappa$ has a subset $D$ that is a club subset of $\kappa$ in the ground model; therefore every stationary subset of $\kappa$ remains stationary in $M[G]$.<br />
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$(\mathbb{P},\leq)$ is ''$\kappa$-distributive'' if the intersection of $\kappa$ open dense sets is still open dense. $\kappa$-distributive notions for infinite $\kappa$ does not add new subsets to $\kappa$. A stronger property, closure, is defined the following way: $\mathbb{P}$ is ''$\kappa$-closed'' if every $\lambda\leq\kappa$, every descending sequence $p_0\geq p_1\geq...\geq p_\alpha\geq... (\alpha<\lambda)$ has a lower bound. Every $\kappa$-closed notion is $\kappa$-distributive. If, for some regular uncountable cardinal $\kappa$ and all $\lambda<\kappa$, $(\mathbb{P},\leq)$ is a $\lambda$-closed forcing notion, then every stationary subset of $\kappa$ remains stationary in every generic extension.<br />
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$(\mathbb{P},\leq)$ has ''property (K)'' (or just ''Knaster property'') if every uncountable set of conditions has an uncountable subet of pairwise compatible elements. Every notion with property (K) satisfies the c.c.c.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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A poset is ''productive-ccc'', if its product with any ccc poset is also ccc (in short $\textit{Prod-ccc}$).<br />
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A ccc poset $\mathbb{P}$ is ''strongly-$\underset{\sim}{Σ_n}$'' if it is $Σ_n$-definable in $H(ω_1)$ with parameters, and the predicate “$x$ codes a maximal antichain of $\mathbb{P}$” is also $Σ_n$-definable in $H(ω_1)$ with parameters.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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A [[projective]] poset $\mathbb{P}$ is ''absolutely-ccc'' if it is ccc in every inner model $W$ of $V$ which satisfies ZFC and contains the parameters of the definition of $\mathbb{P}$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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=== Cohen forcing, adding subsets of regular cardinals, and independence of the continuum hypothesis ===<br />
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Let $\kappa$ be a regular cardinal satisfying $2^{<\kappa}=\kappa$. Let $\lambda>\kappa$ be a cardinal such that $\lambda^\kappa=\lambda$. Let $\text{Add}(\kappa,\lambda) = (\mathbb{P},\leq)$ be the following partial order: $\mathbb{P}$ is the set of all functions $p$ with $\text{dom}(p)\subseteq\lambda\times\kappa$, $|\text{dom}(p)|<\kappa$ and $\text{ran}(p)\subseteq\{0,1\}$, and let $p\leq q$ iff $p\supseteq q$. Let $G$ be a $V$-generic on $\mathbb{P}$ and let $f=\bigcup G$. Then in $V[G]$, $f$ is a function from $\lambda\times\kappa$ to $\{0,1\}$. For every particular $\alpha<\lambda$, the function $c_\alpha(\xi)=f(\alpha,\xi)$ is in $V[G]$ the characteristic function of a subset $x_\alpha=\{\xi<\kappa:c_\alpha(\xi)=1\}$ of $\kappa$. None of those new subsets were originally in $V$, and if $\alpha\neq\beta$ then $x_\alpha\neq x_\beta$. Then, because $\mathbb{P}$ satisfies the $\kappa^+$-chain condition, it follows that all cardinals are preserved except that $2^\kappa=\lambda$.<br />
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In the special case $\kappa=\aleph_0$, there are new real numbers in $V[G]$ and $2^{\aleph_0}=\lambda$. Those new real numbers are called ''Cohen reals''. This technique allows one to show that $\text{ZFC}$ is consistent with the negation of the continuum hypothesis, i.e. that $2^{\aleph_0}>\aleph_1$. In fact, $2^{\aleph_0}$ can be any cardinal with uncountable cofinality, even if singular, e.g. one can force $2^{\aleph_0}=\aleph_{\omega_1}$. Note that $2^{\aleph_0}$ cannot be a cardinal of countable cofinality, so this is impossible to force.<br />
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=== Axiom A ===<br />
A forcing notion $(\mathbb{P},\leq)$ satisfies ''Axiom A'' if there is a sequence of partial orderings $\{\leq_n:n<\omega\}$ of $\mathbb{P}$ such that $p\leq_0 q$ implies $p\leq q$, for all n $p\leq_{n+1} q$ implies $p\leq_n q$, and the following conditions holds:<br />
* for every descending sequence $p_0\geq_0 p_1\geq_1...\geq_{n-1}p_n\geq_n...$ there is a $q$ such that $q\leq_n p_n$ for all $n$.<br />
* for every $p\in\mathbb{P}$, for every $n$ and every ordinal name $\alpha$ there exists $q\leq_n p$ and a countable set $B$ such that $p\Vdash\alpha\in B$.<br />
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Every c.c.c. or $\omega$-closed notion satisfies Axiom A.<br />
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=== Proper forcing ===<br />
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We say that a forcing notion $(\mathbb{P},\leq)$ is ''proper'' if for every uncountable cardinal $\lambda$, every [[stationary]] subset of $[\lambda]^\omega$ remains stationary in every generic extension.<br />
* Every c.c.c. or $\omega$-closed notion is proper, and so is every notion satisfying Axiom A.<br />
* Proper forcing does not collapse $\omega_1$: if $\mathbb{P}$ is proper then every countable set of ordinals in $M[G]$ is a subset of a countable set in $M$.<br />
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''Semi-proper forcing'' ''TODO''<br />
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A projective poset $\mathbb{P}$ is ''strongly-proper'' if for every countable transitive model $N$ of a fragment of ZFC with the parameters of the definition of $\mathbb{P}$ in $N$ and such that $(\mathbb{P}^N, ≤_\mathbb{P}^N, ⊥_\mathbb{P}) ⊆ (\mathbb{P}, ≤_\mathbb{P}, ⊥_\mathbb{P})$, and for every $p ∈ \mathbb{P}^N$, there is $q ≤ p$ which is $(N, \mathbb{P})$-generic, i.e., if $N \models$ “$A$ is a maximal antichain of $\mathbb{P}$”, then $A ∩ N$ is predense below $q$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
* A projective strongly-proper poset is proper.<br />
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=== The Lévy collapse ===<br />
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=== Prikry forcing ===<br />
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=== Other types of forcing, relations ===<br />
(subsection from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>)<br />
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Among important classes of posets are:<br />
* ''$σ$-centered'' posets (can be partitioned into countably many classes so that each class is finite-wise compatible)<br />
* ''$σ$-linked'' posets (can be partitioned into countably many classes so that each class is pair-wise compatible)<br />
* posets that preserve stationary subsets of $ω_1$ (in short $\textit{Stat-pres}$)<br />
* posets that preserve $ω_1$ (in short $\textit{$ω_1$-pres}$)<br />
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We have<br />
: $\textit{$\sigma$-centered} \subset \textit{$\sigma$-linked} \subset \textit{Knaster} \subset \textit{Prod-ccc} \subset \textit{ccc} \subset \textit{Proper} \subset \textit{Semi-proper} \subset \textit{Stat-pres} \subset \textit{$ω_1$-pres}$<br />
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Besides:<br />
* the poset for adding a Cohen real ($\textit{Cohen}$, compare subsection above)<br />
* the poset for adding a random real ($\textit{Random}$)<br />
* the amoeba poset for measure ($\textit{Amoeba}$)<br />
* the amoeba poset for category ($\textit{Amoeba-category}$)<br />
* the Hechler forcing for adding a dominating real<br />
* the Mathias forcing<br />
* Borel forcing notions (the set of conditions is a Borel set)<br />
* the $σ$-linked forcing notion for adding $ω_1$ random reals ($\textit{$ω_1$-Random}$)<br />
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== Product forcing ==<br />
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== Iterated forcing ==<br />
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== Forcing axioms ==<br />
=== Martin's axiom ===<br />
'''Martin's axiom''' ($\text{MA}$) is the following assertion: If $(\mathbb{P},\leq)$ is a forcing notion that satisfies the countable chain condition and if $\mathcal{D}$ is a collection of fewer than $2^{\aleph_0}$ dense subsets of $\mathbb{P}$, then there exists a $\mathcal{D}$-generic filter on $\mathbb{P}$. By replacing "fewer than $2^{\aleph_0}$" by "at most $\kappa$" on obtain the axiom $\text{MA}_\kappa$. Martin's axiom is then $\text{MA}_{<2^{\aleph_0}}$. Note that $\text{MA}_{\aleph_0}$ is provably true in $\text{ZFC}$.<br />
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For all $\kappa$, $\text{MA}_\kappa$ implies $\kappa<2^{\aleph_0}$. Martin's axiom follows from the continuum hypothesis, but is also consistent with its negation. $\text{MA}_{\aleph_1}$ implies there are no [[Suslin tree|Suslin trees]], that every [[tree property|Aronszajn tree]] is special, and that the c.c.c. is equivalent to property (K).<br />
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Martin's axiom implies that $2^{\aleph_0}$ is regular, that it is not real-valued measurable, and also that $2^\lambda=2^{\aleph_0}$ for all $\lambda<2^{\aleph_0}$. It implies that the intersection of fewer than $2^{\aleph_0}$ dense open sets is dense, the union of fewer than $2^{\aleph_0}$ null sets is null, and the union of fewer than $2^{\aleph_0}$ meager sets is meager. Also, the Lebesgue measure is $2^{\aleph_0}$-additive. If additionally $\neg\text{CH}$ then every $\mathbf{\Sigma}^1_2$ set is Lebesgue measurable and has the Baire property.<br />
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=== Proper Forcing Axiom ===<br />
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The '''Proper Forcing Axiom''' ($\text{PFA}$) is obtained by replacing "c.c.c." by "proper" in the definition of $\text{MA}_{\aleph_1}$: for every proper forcing notion $(\mathbb{P},\leq)$, if $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. $\text{PFA}$ implies $2^{\aleph_0}=\aleph_2$ and that the continuum (i.e. $\aleph_2$) has the [[tree property]]. It also implies that every two $\aleph_1$-dense sets of reals are isomorphic.<br />
<br />
Unlike Martin's axiom, which is equiconsistent with $\text{ZFC}$, the $\text{PFA}$ has very high consistency strength, slightly below that of a [[supercompact]] cardinal. If there is a supercompact cardinal then there is a generic extension in which that supercompact is $\aleph_2$ and $\text{PFA}$ holds. On the other hand, [http://www.math.uni-bonn.de/people/pholy/acc_accepted.pdf] proves a ''quasi lower bound'' on the consistency strength of the $\text{PFA}$, which is at least the existence of a proper class of [[subcompact]] cardinals. [https://ac.els-cdn.com/S0001870811002635/1-s2.0-S0001870811002635-main.pdf?_tid=86c2030e-cca4-11e7-b23b-00000aab0f26&acdnat=1511039455_137e37101cda34d46bb0f195cbe43148] also shows that all known methods of forcing $\text{PFA}$ requires a [[strongly compact]] cardinal, and if one wants the forcing to be proper, a supercompact is required.<br />
<br />
$\text{PFA}$ implies the failure of the [[square principle]] $\Box_\kappa$ for every uncountable cardinal $\kappa$, therefore it implies the [[axiom of determinacy|axiom of quasi-projective determinacy]]. It also implies the '''Open Coloring Axiom:''' let $X$ be a set of reals, and let $K\subseteq[X]^2$. We say that $K$ is ''open'' if the set $\{(x,y):\{x,y\}\in K\}$ is an open set in the space $X\times X$. Then<br />
* '''($\text{OCA}$).''' For every $X\subseteq\mathbb{R}$, and for any partition $[X]^2=K_0\cup K_1$ with $K_0$ open, either there exists an uncountable $Y\subseteq X$ such that $[Y]^2\subseteq K_0$ or there exists sets $H_n, n<\omega$ such that $X=\bigcup_{n<\omega}H_n$ and $[H_n]^2\subseteq K_1$ for all $n$.<br />
This axiom has many useful implications in combinatorial set theory.<br />
<br />
Statement equivalent to $\text{PFA}$: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
<br />
=== Martin's maximum and the semiproper forcing axiom ===<br />
<br />
'''Martin's Maximum''' is a strengthening of the proper forcing axiom defined the following way: suppose $(\mathbb{P},\leq)$ is a forcing notion that preserves stationary subsets of $\omega_1$, and that $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. This implies the proper forcing axiom, and also that the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated. It also implies that for all $\kappa\geq\aleph_2$, if $\kappa$ is regular then $\kappa^{\aleph_0}=\kappa$.<br />
<br />
=== Bounded Forcing Axiom ===<br />
&#40;Section from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>)<br />
<br />
The '''Bounded Forcing Axiom''' for a partial ordering $\mathbb{P}$, in short $BFA&#40;\mathbb{P})$, is the statement:<br />
: For every collection $\{I_α : α < ω_1 \}$ of maximal antichains of $\mathbf{B} \overset{def}{=} r.o.&#40;\mathbb{P}) \setminus \{\mathbf{0}\}$, each of size $≤ ω_1$, there is a filter $G ⊆ \mathbf{B}$ such that for every $α$, $I_α ∩ G ≠ ∅$.<br />
For a class of posets $Γ$, $BFA&#40;Γ)$ is the statement that for every $\mathbb{P} ∈ Γ$, $BFA&#40;\mathbb{P})$.<br />
<br />
Examples:<br />
* $MA_{ω_1}$, Martin’s axiom for $ω_1$, is $BFA&#40;\textit{ccc})$.<br />
* $BPFA$, the bounded proper forcing axiom, is $BFA&#40;\textit{Proper})$.<br />
* $BSPFA$, the bounded semi-proper forcing axiom, is $BFA&#40;\textit{Semi-proper})$.<br />
* $BMM$, the bounded Martin’s maximum, is $BFA&#40;\textit{Stat-pres})$.<br />
<br />
$BPFA$ implies that there is a well-ordering of the reals in length $ω_2$ definable with parameters in $H&#40;ω_2)$ and therefore $\mathfrak{c} = \aleph_2$.<br />
<br />
$BMM$ implies that for every set $X$ there is an inner model with a [[strong]] cardinal containing $X$.<br />
* Thus, in particular, $BMM$ implies that for every set $X$, [[zero dagger|$X^\dagger$ exists]].<br />
<br />
=== $\text{wPFA}$ and $\text{PFA}_κ$ ===<br />
(information in this subsection from <cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite>)<br />
<br />
The '''weak Proper Forcing Axiom''' is obtained by requiring only that embedding $j$ (like in the last statement equivalent to $\text{PFA}$) exists in a forcing extension: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ in $V$ and an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ in a set-forcing extension (equivalently in $V^{Coll(ω, \bar{M})}$) such that $φ(\bar{\mathcal{M}})$ holds.<br />
<br />
If there is a [[remarkable]] cardinal, then $\text{wPFA}$ holds in a forcing extension by a proper poset. If $\text{wPFA}$ holds, then $ω_2^V$ is remarkable in $L$.<br />
<br />
For a cardinal $κ$, $\text{PFA}_κ$ is the statement that<br />
: if $\mathbb{B}$ is any proper complete Boolean algebra and if $\langle A_ξ | ξ < ω_1 \rangle$ is any family of maximal antichains in $\mathbb{B}$ with $|A_ξ| ≤ κ$ for each $ξ < ω_1$, then there is some filter $G ⊆ \mathbb{B}$ such that $\forall_{ξ < ω_1} G ∩ A_ξ ≠ ∅$.<br />
Equivalently, in analogy to the other statements (adding the assumption $|M| ≤ κ$):<br />
: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $|M| ≤ κ$, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.<br />
<br />
$\text{PFA}_{\aleph_1}$ is $\text{BPFA}$.<br />
<br />
$\text{wPFA}$ implies $\text{PFA}_{\aleph_2}$. However, it does not imply $\text{PFA}_{\aleph_3}$, because assertion $\text{wPFA} ∧ ∀_{κ ≥ \aleph_2} \square_κ$ is consistent relative to a remarkable cardinal and (Todorcevic, 1984, Theorem 1) $\text{PFA}_{\aleph_3}$ implies the failure of $\square_{ω_2}$.<br />
<br />
== Generic absoluteness and universal Baireness ==<br />
<br />
''Main article: [[Universally Baire]]''<br />
<br />
== Generic ultrapowers ==<br />
<br />
''Main article: [[Filter#Precipitous ideals|Precipitous ideals]]''<br />
<br />
== Axioms of generic absoluteness ==<br />
<br />
''Main article: [[Axioms of generic absoluteness]]''<br />
<br />
{{stub}}<br />
<br />
{{references}}<br />
<br />
[[Category:Forcing]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Superstrong&diff=4174Superstrong2022-05-29T13:32:16Z<p>BartekChom: /* Relation to other large cardinal notions */ indestructible</p>
<hr />
<div>[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for the [[axiom of determinacy]]. However, Shelah had then discovered that [[Shelah]] cardinals were a weaker bound that still sufficed to imply the consistency strength of $\text{(ZF+)AD}$. After this, it was found that the existence of infinitely many [[Woodin]] cardinals was equiconsistent to $\text{AD}$. Woodin-ness is a significant weakening of superstrongness.<br />
<br />
''Most results in this article can be found in <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
There are, like all critical point variations on [[measurable]] cardinals, multiple equivalent definitions of superstrongness. In particular, there is an [[elementary embedding]] definition and an [[extender]] definition.<br />
<br />
=== Elementary Embedding Definition ===<br />
<br />
A cardinal $\kappa$ is '''$n$-superstrong''' (or $n$-fold superstrong when referring to the [[n-fold variants|$n$-fold variants]]) iff it is the critical point of some [[elementary embedding]] $j:V\rightarrow M$ such that $M$ is a transitive class and $V_{j^n(\kappa)}\subset M$ (in this case, $j^{n+1}(\kappa):=j(j^n(\kappa))$ and $j^0(\kappa):=\kappa$).<br />
<br />
A cardinal is '''superstrong''' iff it is $1$-superstrong.<br />
<br />
The definition quite clearly shows that $\kappa$ is $j^n(\kappa)$-[[strong]]. However, the least superstrong cardinal is never strong.<br />
<br />
$j^k(\kappa)$ for $1 \le k \le n$ are called targets.<cite>Kentaro2007:DoubleHelix</cite><br />
<br />
=== Extender Definition ===<br />
<br />
A cardinal $\kappa$ is '''$n$-superstrong''' (or $n$-fold superstrong) iff there is a [[extender|$(\kappa,\beta)$-extender]] $\mathcal{E}$ for a $\beta>\kappa$ with $V_{j^n_{\mathcal{E}}(\kappa)}\subseteq$ [[ultrapower|$Ult_{\mathcal{E}}(V)$]] (where $j_{\mathcal{E}}$ is the canonical ultrapower embedding from $V$ into $Ult_{\mathcal{E}}(V)$).<br />
<br />
A cardinal is '''superstrong''' iff it is $1$-superstrong.<br />
<br />
== Relation to other large cardinal notions ==<br />
<br />
The consistency strength of $n$-superstrongness follows the [[n-fold variants|double helix pattern]] <cite>Kentaro2007:DoubleHelix</cite>. Specifically:<br />
*[[measurable]] = $0$-superstrong = [[huge|almost $0$-huge]] = $0$-huge<br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $&#40;n+1)$-fold strong, $n$-fold extendible<br />
* $&#40;n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $&#40;n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $&#40;n+1)$-superstrong<br />
<br />
Let $M$ be a transitive class $M$ such that there exists an elementary embedding $j:V\to M$ with $V_{j&#40;\kappa)}\subseteq M$, and let $\kappa$ be its superstrong critical point. While $j&#40;\kappa)$ need not be an inaccessible cardinal in $V$, it is always [[worldly]] and the rank model $V_{j&#40;\kappa)}$ satisfies $\text{ZFC+}$"$\kappa$ is strong" &#40;although $\kappa$ may not be strong in $V$).<br />
<br />
Superstrong cardinals have strong upward reflection properties, in particular there are many [[measurable]] cardinals ''above'' a superstrong cardinal. Every $n$-huge cardinal is $n$-superstrong, and so $n$-huge cardinals also have strong reflection properties. Remark however that if $\kappa$ is [[strong]] or [[supercompact]], then it is consistent that there is no inaccessible cardinals larger than $\kappa$: this is because if $\lambda>\kappa$ is inaccessible, then $V_\lambda$ satisfies $\kappa$'s strongness/supercompactness. Thus it is clear that supercompact cardinals need not be superstrong, even though they have higher consistency strength. In fact, because of the downward reflection properties of strong/supercompact cardinals, if there is a superstrong above a strong/supercompact $\kappa$, then there are $\kappa$-many superstrong cardinals below $\kappa$; same with hugeness instead of superstrongness. In particular, the least superstrong is strictly smaller than the least strong &#40;and thus smaller than the least supercompact).<br />
<br />
* $n$-fold supercompactness implies $n$-fold “super”-superstrongness &#40;$n$-superstrongness with arbitrary large first target).<cite>Kentaro2007:DoubleHelix</cite><br />
* Every [[extendible|$1$-extendible]] cardinal is superstrong and has a [[filter|normal measure]] containing all of the superstrongs less than said $1$-extendible. This means that the set of all superstronges less than it is [[stationary]]. Similarly, every cardinal $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]] is larger than the least superstrong cardinal and has a normal measure containing all of the superstrongs less than it.<br />
* Every superstrong cardinal is [[Woodin]] and has a normal measure containing all of the Woodin cardinals less than it. Thus the set of all Woodin cardinals below it is stationary, and so is the set of all measurables smaller than it.<br />
* If $κ$ is superstrong, then it is [[Shelah]] and there are $κ$ Shelah cardinals below it.<cite>Golshani2017:EastonLikeInPresenceShelah</cite><br />
* Superstrongness is consistency-wise stronger than [[Woodin|hyper-Woodinness]].<cite>Schimmerling2002:WoodinShelahAndCoreModel</cite><br />
* If there is a superstrong cardinal, then in [[Constructible universe|$L&#40;\mathbb{R})$]], the [[axiom of determinacy]] holds. <cite>Jech2003:SetTheory</cite><br />
* Letting $\kappa$ be superstrong, $\kappa$ can be [[forcing|forced]] to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in this forcing extension there is a normal $\omega_2$-saturated ideal on $\omega_1$. <cite>Jech2003:SetTheory</cite><br />
* Superstrongness is not Laver [[indestructible]]. <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite> <br />
* If [[I4|$\mathrm{I}_4^1&#40;\kappa)$]], then $\kappa$ is superstrong and $\{\alpha\lt\kappa|\alpha\text{ is superstrong}\}$ has measure 1.<cite>Corazza2003:GapBetweenI3andWA</cite><br />
<br />
A cardinal $κ$ is '''[[correct|$C^{&#40;n)}$-superstrong]]''' iff there exists an elementary embedding $j : V → M$ for transitive $M$, with $crit&#40;j) = κ$, $V_{j&#40;κ)} ⊆ M$ and $j&#40;κ) ∈ C^{&#40;n)}$.<cite>Bagaria2012:CnCardinals</cite><br />
* Every $C^{&#40;n)}$-superstrong cardinal belongs to $C^{&#40;n)}$.<br />
* Every superstrong cardinal is $C^{&#40;1)}$-superstrong.<br />
* For every $n ≥ 1$, if $κ$ is $C^{&#40;n+1)}$-superstrong, then there is a $κ$-complete normal [[ultrafilter]] $U$ over $κ$ such that $\{α < κ : α$ is $C^{&#40;n)}$-superstrong$\} ∈ U$. Hence, the first $C^{&#40;n)}$-superstrong cardinal, if it exists, is not $C^{&#40;n+1)}$-superstrong.<br />
* If $κ$ is $2^κ$-[[supercompact]] and belongs to $C^{&#40;n)}$, then there is a $κ$-complete normal ultrafilter $U$ over $κ$ such that the set of $C^{&#40;n)}$-superstrong cardinals smaller than $κ$ belongs to $U$.<br />
* If $κ$ is $κ+1$-$C^{&#40;n)}$-[[extendible]] and belongs to $C^{&#40;n)}$, then $κ$ is $C^{&#40;n)}$-superstrong and there is a $κ$-complete normal ultrafilter $U$ over $κ$ such that the set of $C^{&#40;n)}$-superstrong cardinals smaller than $κ$ belongs to $U$.<br />
* Every $C^{&#40;n)}$-[[huge|almost-huge]] cardinal is $C^{&#40;n)}$-superstrong.<br />
* Assuming [[rank into rank|$\mathrm{I3}&#40;κ, δ)$]], if $δ$ is a limit cardinal &#40;instead of a successor of a limit cardinal – Kunen’s Theorem excludes other cases), it is equal to $sup\{j^m&#40;κ) : m ∈ ω\}$ where $j$ is the elementary embedding. Then $κ$ and $j^m&#40;κ)$ are $C^{&#40;n)}$-superstrong &#40;inter alia) in $V_δ$, for all $n$ and $m$.<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Uplifting&diff=4173Uplifting2022-05-29T13:31:28Z<p>BartekChom: /* Weakly superstrong cardinal */ indestructible</p>
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<div>{{DISPLAYTITLE: Uplifting cardinals}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
<br />
Uplifting cardinals were introduced by Hamkins and Johnstone in <cite>HamkinsJohnstone:ResurrectionAxioms</cite>, from which some of this text is adapted.<br />
<br />
An [[inaccessible]] cardinal $\kappa$ is '''uplifting''' if and only if for every ordinal $\theta$ it is '''$\theta$-uplifting''', meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension. <br />
<br />
An inaccessible cardinal is '''pseudo uplifting''' if and only if for every ordinal $\theta$ it is '''pseudo $\theta$-uplifting''', meaning that there is a cardinal $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension, without insisting that $\gamma$ is inaccessible.<br />
<br />
Being '''strongly uplifting''' (see further) is boldface variant of being uplifting.<br />
<br />
It is an elementary exercise to see that if $V_\kappa\prec V_\gamma$ is a proper elementary extension, then $\kappa$ and hence also $\gamma$ are [[Beth fixed point | $\beth$-fixed points]], and so $V_\kappa=H_\kappa$ and $V_\gamma=H_\gamma$. It follows that a cardinal $\kappa$ is uplifting if and only if it is regular and there are arbitrarily large regular cardinals $\gamma$ such that $H_\kappa\prec H_\gamma$. It is also easy to see that every uplifting cardinal $\kappa$ is uplifting in $L$, with the same targets. Namely, if $V_\kappa\prec V_\gamma$, then we may simply restrict to the constructible sets to obtain $V_\kappa^L=L^{V_\kappa}\prec L^{V_\gamma}=V_\gamma^L$. An analogous result holds for pseudo-uplifting cardinals.<br />
<br />
== Consistency strength of uplifting cardinals == <br />
<br />
The consistency strength of uplifting and pseudo-uplifting cardinals are bounded between the existence of a [[Mahlo]] cardinal and the hypothesis [[Ord is Mahlo]]. <br />
<br />
'''Theorem.''' <br />
<br />
1. If $\delta$ is a [[Mahlo]] cardinal, then $V_\delta$ has a proper class of uplifting cardinals.<br />
<br />
2. Every uplifting cardinal is pseudo uplifting and a limit of pseudo uplifting cardinals.<br />
<br />
3. If there is a pseudo uplifting cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a [[reflecting]] cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. <br />
<br />
Proof. For (1), suppose that $\delta$ is a Mahlo cardinal. By the Lowenheim-Skolem theorem, there is a club set $C\subset\delta$ of cardinals $\beta$ with $V_\beta\prec V_\delta$. Since $\delta$ is Mahlo, the club $C$ contains unboundedly many inaccessible cardinals. If $\kappa<\gamma$ are both in $C$, then $V_\kappa\prec V_\gamma$, as desired. Similarly, for (2), if $\kappa$ is uplifting, then $\kappa$ is pseudo uplifting and if $V_\kappa\prec V_\gamma$ with $\gamma$ inaccessible, then there are unboundedly many ordinals $\beta<\gamma$ with $V_\beta\prec V_\gamma$ and hence $V_\kappa\prec V_\beta$. So $\kappa$ is pseudo uplifting in $V_\gamma$. From this, it follows that there must be unboundedly many pseudo uplifting cardinals below $\kappa$. For (3), if $\kappa$ is inaccessible and $V_\kappa\prec V_\gamma$, then $V_\gamma$ is a transitive set model of ZFC in which $\kappa$ is reflecting, and it is thus also a model of [[Ord is Mahlo]]. QED<br />
<br />
== Uplifting cardinals and $\Sigma_3$-reflection == <br />
<br />
* Every uplifting cardinal is a limit of $\Sigma_3$-reflecting cardinals, and is itself $\Sigma_3$-reflecting.<br />
* If $\kappa$ is the least uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
The analogous observation for pseudo uplifting cardinals holds as well, namely, every pseudo uplifting cardinal is $\Sigma_3$-reflecting and a limit of $\Sigma_3$-reflecting cardinals; and if $\kappa$ is the least pseudo uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
== Uplifting Laver functions ==<br />
<br />
Every uplifting cardinal admits an ordinal-anticipating Laver function, and indeed, a HOD-anticipating Laver function, a function $\ell:\kappa\to V_\kappa$, definable in $V_\kappa$, such that for any set $x\in\text{HOD}$ and $\theta$, there is an inaccessible cardinal $\gamma$ above $\theta$ such that $V_\kappa\prec V_\gamma$, for which $\ell^*(\kappa)=x$, where $\ell^*$ is the corresponding function defined in $V_\gamma$. <br />
<br />
== Connection with the resurrection axioms ==<br />
<br />
Many instances of the (weak) resurrection axiom imply that ${\frak c}^V$ is an uplifting cardinal in $L$:<br />
* RA(all) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* RA(ccc) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* wRA(countably closed)+$\neg$CH implies that ${\frak c}^V$ is uplifting in $L$.<br />
* Under $\neg$CH, the weak resurrection axioms for the classes of axiom-A forcing, proper forcing, semi-proper forcing, and posets that preserve stationary subsets of $\omega_1$, respectively, each imply that ${\frak c}^V$ is uplifting in $L$.<br />
<br />
Conversely, if $\kappa$ is uplifting, then various resurrection axioms hold in a corresponding lottery-iteration forcing extension. <br />
<br />
'''Theorem.''' (Hamkins and Johnstone) The following theories are equiconsistent over ZFC:<br />
* There is an uplifting cardinal.<br />
* RA(all)<br />
* RA(ccc)<br />
* RA(semiproper)+$\neg$CH<br />
* RA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, RA($\alpha$-proper)+$\neg$CH<br />
* RA(axiom-A)+$\neg$CH<br />
* wRA(semiproper)+$\neg$CH<br />
* wRA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, wRA($\alpha$-proper})+$\neg$CH<br />
* wRA(axiom-A)+$\neg$CH<br />
* wRA(countably closed)+$\neg$CH<br />
<br />
== Strongly Uplifting ==<br />
<br />
(Information in this section comes from <cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite>)<br />
<br />
Strongly uplifting cardinals are precisely strongly pseudo uplifting ordinals, strongly uplifting cardinals with weakly compact targets, superstrongly [[unfoldable]] cardinals and almost-hugely unfoldable cardinals.<br />
<br />
=== Definitions ===<br />
<br />
An ordinal is '''strongly pseudo uplifting''' iff for every ordinal $θ$ it is '''strongly $θ$-uplifting''', meaning that for every $A⊆V_κ$, there exists some ordinal $λ>θ$ and an $A^*⊆V_λ$ such that $(V_κ;∈,A)≺(V_λ;∈,A^*)$ is a proper elementary extension.<br />
<br />
An inaccessible cardinal is '''strongly uplifting''' iff for every ordinal $θ$ it is '''strongly $θ$-uplifting''', meaning that for every $A⊆V_κ$, there exists some inaccessible(*) $λ>θ$ and an $A^*⊆V_λ$ such that $(V_κ;∈,A)≺(V_λ;∈,A^*)$ is a proper elementary extension. By replacing starred "inaccessible" with "weakly compact" and other properties, we get strongly uplifting with weakly compact etc. targets.<br />
<br />
A cardinal $\kappa$ is '''$\theta$-superstrongly unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subseteq N$.<br />
<br />
A cardinal $\kappa$ is '''$\theta$-almost-hugely unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $N^{<j(\kappa)}\subseteq N$.<br />
<br />
$κ$ is then called '''superstrongly unfoldable''' (resp. '''almost-hugely unfoldable''') iff it is $θ$-strongly unfoldable (resp. $θ$-almost-hugely unfoldable) for every $θ$; i.e. the target of the embedding can be made arbitrarily large.<br />
<br />
=== Equivalence ===<br />
For any ordinals $κ$, $θ$, the following are equivalent:<br />
* $κ$ is strongly pseudo $(θ+1)$-uplifting.<br />
* $κ$ is strongly $(θ+1)$-uplifting.<br />
* $κ$ is strongly $(θ+1)$-uplifting with weakly compact targets.<br />
* $κ$ is strongly $(θ+1)$-uplifting with totally indescribable targets, and indeed with targets having any property of $κ$ that is absolute to all models $V_γ$ with $γ > κ, θ$.<br />
<br />
For any cardinal $κ$ and ordinal $θ$, the following are equivalent:<br />
* $κ$ is strongly $(θ+1)$-uplifting.<br />
* $κ$ is superstrongly $(θ+1)$-unfoldable.<br />
* $κ$ is almost-hugely $(θ+1)$-unfoldable.<br />
* For every set $A ∈ H_{κ^+}$ there is a $κ$-[[model]] $M⊨\mathrm{ZFC}$ with $A∈M$ and $V_κ≺M$ and a transitive set $N$ with an elementary embedding $j:M→N$ having critical point $κ$ with $j(κ)> θ$ and $V_{j(κ)}≺N$, such that $N^{<j(κ)}⊆N$ and $j(κ)$ is inaccessible, weakly compact and more in $V$.<br />
* $κ^{<κ}=κ$ holds, and for every $κ$-model $M$ there is an elementary embedding $j:M→N$ having critical point $κ$ with $j(κ)> θ$ and $V_{j(κ)}⊆N$, such that $N^{<j(κ)}⊆N$ and $j(κ)$ is inaccessible, weakly compact and more in $V$.<br />
<br />
=== Relations to other cardinals ===<br />
* If $δ$ is a subtle cardinal, then the set of cardinals $κ$ below $δ$ that are strongly uplifting in $V_δ$ is stationary.<br />
* If $0^♯$ exists, then every Silver indiscernible is strongly uplifting in $L$.<br />
* In $L$, $κ$ is strongly uplifting iff it is unfoldable with cardinal targets.<br />
* Every strongly uplifting cardinal is strongly uplifting in $L$. Every strongly $θ$-uplifting cardinal is strongly $θ$-uplifting in $L$.<br />
* Every strongly uplifting cardinal is strongly unfoldable of every ordinal degree $α$ and a stationary limit of cardinals that are strongly unfoldable of every ordinal degree and so on.<br />
<br />
=== Relation to boldface resurrection axiom ===<br />
The following theories are equiconsistent over $\mathrm{ZFC}$:<br />
* There is a strongly uplifting cardinal.<br />
* The boldface resurrection axiom for all forcing, for proper forcing, for semi-proper forcing and for c.c.c. forcing.<br />
* The weak boldface resurrection axioms for countably-closed forcing, for axiom-$A$ forcing, for proper forcing and for semi-properforcing, respectively, plus $¬\mathrm{CH}$.<br />
<br />
== Weakly superstrong cardinal ==<br />
(Information in this section comes from <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite>)<br />
<br />
Hamkins and Johnstone called an inaccessible cardinal $κ$ '''weakly superstrong''' if for every transitive set $M$ of size $κ$ with $κ∈M$ and $M^{<κ}⊆M$, a transitive set $N$ and an elementary embedding $j:M→N$ with critical point $κ$, for which $V_{j(κ)}⊆N$, exist.<br />
<br />
It is called '''weakly almost huge''' if for every such $M$ there is such $j:M→N$ for which $N^{<j(κ)}⊆N$.<br />
<br />
(As usual one can call $j(κ)$ the target.)<br />
<br />
A cardinal is superstrongly unfoldable if it is weakly superstrong with arbitrarily large targets, and it is almost hugely unfoldable if it is weakly almost huge with arbitrarily large targets.<br />
<br />
If $κ$ is weakly superstrong, it is $0$-[[extendible]] and $\Sigma_3$-[[extendible]]. Weakly almost huge cardinals also are $\Sigma_3$-[[extendible]]. Because $\Sigma_3$-extendibility always can be destroyed, all these cardinal properties (among others) are never Lever [[indestructible]].<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Tall&diff=4172Tall2022-05-29T13:30:19Z<p>BartekChom: /* Tall Cardinals */ indestructible</p>
<hr />
<div>{{DISPLAYTITLE: Tall cardinal}}[[Category:Large cardinal axioms]]<br />
<br />
== Tall Cardinals ==<br />
<br />
A cardinal $\kappa$ is '''$\theta$-tall''' iff there is an [[elementary embedding]] $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j&#40;\kappa)>\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j&#40;\kappa)$ can be made arbitrarily large. Every [[strong]] cardinal is tall and every [[strongly compact]] cardinal is tall, but [[measurable]] cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a [[strong]] cardinal. Any tall cardinal $\kappa$ can be made [[indestructible]] by a variety of forcing notions,<br />
including forcing that pumps up the value of $2^\kappa$ as high as desired. See <cite>Hamkins2009:TallCardinals</cite>.<br />
<br />
=== Extender Characterization ===<br />
<br />
If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $&#40;\kappa,\theta^+)$-extender $E$ such that, if $M\cong Ult_E$ is the ultrapower of $V$ by $E$, $M^\kappa\subset M$. Similarly, $\kappa$ is tall iff for any $\lambda$ there exists some $&#40;\kappa,\lambda)$-extender such that $M^\kappa\subset M$ where $M$ is as above.<br />
<br />
== Strongly Tall Cardinals ==<br />
<br />
A cardinal $\kappa$ is '''strongly $\theta$-tall''' iff there is some [[filter|measure]] $U$ on a set $S$ witnessing $\kappa$'s $\theta$-tallness in the ultrapower of $V$ by $U$. More precisely, the ultrapower embedding $j:V\prec M$ has critical point $\kappa$, $M^\kappa\subset M$, and $j(\kappa)>\theta$. $\kappa$ is '''strongly tall''' iff it is strongly $\theta$-tall for every $\theta$. <br />
<br />
The existence of a strongly tall cardinal is equiconsistent to the existence of a strong cardinal with a proper class of measurables above it (below the consistency strength of a [[Woodin]] cardinal, above the consistency strength of a [[strong]] cardinal and therefore above a tall cardinal). Specifically, if $κ$ is strong and has a proper class of measurables above it and [[continuum hypothesis | GCH]] holds, then in a forcing extension of $V$, $κ$ is strongly tall. On the other hand, if $κ$ is strongly tall and there is no inner model with two strong cardinals, then $κ$ is strong in $K$ and has a proper class of measurables above it in $K$ ($K$ being the [[core model]]).<br />
<br />
=== Ultrapower Characterization ===<br />
<br />
$\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and a $\kappa$-complete [[filter|ultrafilter]] $U$ on $S$ such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)>\theta$. That is, there is an ultrapower of an ultrafilter which witnesses the $\gamma$-tallness of $\kappa$.<br />
<br />
=== Embedding Characterization ===<br />
<br />
If $\theta\geq\kappa$, then $\kappa$ is strongly $\theta$-tall iff $\kappa$ is the critical point of some $j:V\prec M$ for which there is a set $S$ and an $A\in j(S)$ such that for any $\alpha\leq\theta$, there is a function $f:S\rightarrow\kappa$ with $j(f)(A)=\alpha$.<br />
<br />
=== Ultrafilter Characterization ===<br />
<br />
$\kappa$ is strongly $\theta$-tall iff there is some set $S$, a $\kappa$-complete [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow V$ for each ordinal $\alpha$ such that:<br />
#$\kappa$ is uncountable.<br />
#$H_0(x)=0$ for each $x\in S$.<br />
#For each $\alpha$ and each $f:S\rightarrow V$, $\{x\in S:f(x)\in H_\alpha(x)\}\in U$ iff there is some $\beta<\alpha$ such that $\{x\in S:f(x)=H_\beta(x)\}\in U$. That is, $f(x)\in H_\alpha(x)$ almost everywhere iff there is some $\beta<\alpha$ such that $f(x)=H_\beta(x)$ almost everywhere.<br />
#$\{x\in S:H_\theta(x)\in\kappa\}\in U$. That is, $H_\theta(x)\in\kappa$ almost everywhere.<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Extendible&diff=4171Extendible2022-05-29T13:28:49Z<p>BartekChom: /* $\Sigma_n$-extendible cardinals */ indestructible</p>
<hr />
<div>{{DISPLAYTITLE: Extendible cardinal}}<br />
[[Category:Reflection principles]]<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
A cardinal $\kappa$ is ''$\eta$-extendible'' for an ordinal $\eta$ if and only if there is an [[elementary embedding]] $j:V_{\kappa+\eta}\to V_\theta$, with critical point $\kappa$, for some ordinal $\theta$. The cardinal $\kappa$ is ''extendible'' if and only if it is $\eta$-extendible for every ordinal $\eta$. Equivalently, for every ordinal $\alpha$ there is a nontrivial elementary embedding $j:V_{\kappa+\alpha+1}\to V_{j(\kappa)+j(\alpha)+1}$ with critical point $\kappa$.<br />
<br />
== Alternative definition ==<br />
<br />
Given cardinals $\lambda$ and $\theta$, a cardinal $\kappa\leq\lambda,\theta$ is ''jointly $\lambda$-supercompact and $\theta$-superstrong'' if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{crit}(j)=\kappa$, $\lambda<j(\kappa)$, $M^\lambda\subseteq M$ and $V_{j(\theta)}\subseteq M$. That is, a single embedding witnesses both $\lambda$-[[supercompact|supercompactness]] and (a strengthening of) [[superstrong|superstrongness]] of $\kappa$. The least supercompact is never jointly $\lambda$-supercompact and $\theta$-superstrong for any $\lambda$,$\theta\geq\kappa$.<br />
<br />
A cardinal is extendible if and only if it is jointly supercompact and $\kappa$-superstrong, i.e. for every $\lambda\geq\kappa$ it is jointly $\lambda$-supercompact and $\kappa$-superstrong. [http://logicatorino.altervista.org/slides/150619tsaprounis.pdf] One can show that extendibility of $\kappa$ is in fact equivalent to "for all $\lambda$,$\theta\geq\kappa$, $\kappa$ is jointly $\lambda$-supercompact and $\theta$-superstrong". A similar characterization of $C^{(n)}$-extendible cardinals exists.<br />
<br />
The [[huge|ultrahuge]] cardinals are defined in a way very similar to this, and one can (very informally) say that "ultrahuge cardinals are to superhuges what extendibles are to supercompacts". These cardinals are superhuge (and stationary limits of superhuges) and strictly below almost 2-huges in consistency strength.<br />
<br />
''To be expanded: Extendibility Laver Functions''<br />
<br />
== Relation to Other Large Cardinals ==<br />
<br />
Extendible cardinals are related to various kinds of measurable cardinals.<br />
<br />
Hyper-[[huge]] cardinals are extendible limits of extendible cardinals.<cite>Usuba2018:ExtendibleCardinalsAndTheMantle</cite><br />
<br />
The relationship between extendible, [[hypercompact]] and [[Supercompact#Enhanced supercompact cardinals|enhanced supercompact]] cardinals is not known. They all lay between [[supercompact]] and [[Vopenka|Vopěnka]]<cite>Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge</cite>.<br />
<br />
=== Supercompactness ===<br />
<br />
Extendibility is connected in strength with [[supercompact|supercompactness]]. Every extendible cardinal is supercompact, since from the embeddings $j:V_\lambda\to V_\theta$ we may extract the induced supercompactness measures $X\in\mu\iff j''\delta\in j(X)$ for $X\subset \mathcal{P}_\kappa(\delta)$, provided that $j(\kappa)\gt\delta$ and $\mathcal{P}_\kappa(\delta)\subset V_\lambda$, which one can arrange. On the other hand, if $\kappa$ is $\theta$-supercompact, witnessed by $j:V\to M$, then $\kappa$ is $\delta$-extendible inside $M$, provided $\beth_\delta\leq\theta$, since the restricted elementary embedding $j\upharpoonright V_\delta:V_\delta\to j(V_\delta)=M_{j(\delta)}$ has size at most $\theta$ and is therefore in $M$, witnessing $\delta$-extendibility there. <br />
<br />
Although extendibility itself is stronger and larger than [[supercompact|supercompactness]], $\eta$-supercompacteness is not necessarily too much weaker than $\eta$-extendibility. For example, if a cardinal $\kappa$ is $\beth_{\eta}(\kappa)$-supercompact (in this case, the same as $\beth_{\kappa+\eta}$-supercompact) for some $\eta<\kappa$, then there is a normal measure $U$ over $\kappa$ such that $\{\lambda<\kappa:\lambda\text{ is }\eta\text{-extendible}\}\in U$.<br />
<br />
=== Strong Compactness ===<br />
<br />
Interestingly, extendibility is also related to [[strongly compact|strong compactness]]. A cardinal $\kappa$ is strongly compact iff the [[infinitary logic|infinitary language]] $\mathcal{L}_{\kappa,\kappa}$ has the $\kappa$-compactness property. A cardinal $\kappa$ is extendible iff the infinitary language $\mathcal{L}_{\kappa,\kappa}^n$ (the infinitary language but with $(n+1)$-th order logic) has the $\kappa$-compactness property for every natural number $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Given a logic $\mathcal{L}$, the minimum cardinal $\kappa$ such that $\mathcal{L}$ satisfies the $\kappa$-compactness theorem is called the '''strong compactness cardinal''' of $\mathcal{L}$. The strong compactness cardinal of $\omega$-th order finitary logic (that is, the union of all $\mathcal{L}_{\omega,\omega}^n$ for natural $n$) is the least extendible cardinal.<br />
<br />
== Variants ==<br />
<br />
=== $C^{(n)}$-extendible cardinals ===<br />
(Information in this subsection from <cite>Bagaria2012:CnCardinals</cite> unless noted otherwise)<br />
<br />
A cardinal $κ$ is called '''$C^{(n)}$-extendible''' if for all $λ > κ$ it is $λ$-$C^{(n)}$-extendible, i.e. if there is an ordinal $µ$ and an elementary embedding $j : V_λ → V_µ$, with $\mathrm{crit(j)} = κ$, $j(κ) > λ$ and $j(κ) ∈ C^{(n)}$.<br />
<br />
For $λ ∈ C^{(n)}$, a cardinal $κ$ is $λ$-$C^{(n)+}$-extendible iff it is $λ$-$C^{(n)}$-extendible, witnessed by some $j : V_λ → V_µ$ which (besides $j(κ) > λ$ and $j(κ) ∈ C(n)$) satisfies that $µ ∈ C^{(n)}$.<br />
<br />
$κ$ is $C^{(n)+}$-extendible iff it is $λ$-$C^{(n)+}$-extendible for every $λ > κ$ such that $λ ∈ C^{(n)}$.<br />
<br />
Properties:<br />
* The notions of $C^{(n)}$-extendible cardinals and $C^{(n)+}$-extendible cardinals are equivalent.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* Every extendible cardinal is $C^{(1)}$-extendible.<br />
* If $κ$ is $C^{(n)}$-extendible, then $κ ∈ C^{(n+2)}$.<br />
* For every $n ≥ 1$, if $κ$ is $C^{(n)}$-extendible and $κ+1$-$C^{(n+1)}$-extendible, then the set of $C^{(n)}$-extendible cardinals is unbounded below $κ$.<br />
** Hence, the first $C^{(n)}$-extendible cardinal $κ$, if it exists, is not $κ+1$-$C^{(n+1)}$-extendible.<br />
** In particular, the first extendible cardinal $κ$ is not $κ+1$-$C^{(2)}$-extendible.<br />
* For every $n$, if there exists a $C^{(n+2)}$-extendible cardinal, then there exist a proper class of $C^{(n)}$-extendible cardinals.<br />
* The existence of a $C^{(n+1)}$-extendible cardinal $κ$ (for $n ≥ 1$) does not imply the existence of a $C^{(n)}$-extendible cardinal greater than $κ$. For if $λ$ is such a cardinal, then $V_λ \models$“κ is $C^{(n+1)}$-extendible”.<br />
* If $κ$ is $κ+1$-$C^{(n)}$-extendible and belongs to $C^{(n)}$, then $κ$ is $C^{(n)}$-[[superstrong]] and there is a $κ$-complete normal [[ultrafilter]] $U$ over $κ$ such that the set of $C^{(n)}$-superstrong cardinals smaller than $κ$ belongs to $U$.<br />
* For $n ≥ 1$, the following are equivalent ($VP$ — [[Vopenka|Vopěnka's principle]]):<br />
** $VP(Π_{n+1})$<br />
** $VP(κ, \mathbf{Σ_{n+2}})$ for some $κ$<br />
** There exists a $C(n)$-extendible cardinal.<br />
* “For every $n$ there exists a $C(n)$-extendible cardinal.” is equivalent to the full Vopěnka's principle.<br />
* Every $C^{(n)}$-[[huge|superhuge]] cardinal is $C^{(n)}$-extendible.<br />
* Assuming [[rank into rank|$\mathrm{I3}(κ, δ)$]], if $δ$ is a limit cardinal (instead of a successor of a limit cardinal – Kunen’s Theorem excludes other cases), it is equal to $sup\{j^m(κ) : m ∈ ω\}$ where $j$ is the elementary embedding. Then $κ$ and $j^m(κ)$ are $C^{(n)}$-extendible (inter alia) in $V_δ$, for all $n$ and $m$.<br />
<br />
=== $(\Sigma_n,\eta)$-extendible cardinals ===<br />
<br />
There are some variants of extendible cardinals because of the interesting jump in consistency strength from $0$-extendible cardinals to $1$-extendibles. These variants specify the elementarity of the embedding.<br />
<br />
A cardinal $\kappa$ is $(\Sigma_n,\eta)$-extendible, if there is a $\Sigma_n$-elementary embedding $j:V_{\kappa+\eta}\to V_\theta$ with critical point $\kappa$, for some ordinal $\theta$. These cardinals were introduced by Bagaria, Hamkins, Tsaprounis and Usuba <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite>. <br />
<br />
=== $\Sigma_n$-extendible cardinals === <br />
<br />
The special case of $\eta=0$ leads to a much weaker notion. Specifically, a cardinal $\kappa$ is ''$\Sigma_n$-extendible'' if it is $(\Sigma_n,0)$-extendible, or more simply, if $V_\kappa\prec_{\Sigma_n} V_\theta$ for some ordinal $\theta$. Note that this does not necessarily imply that $\kappa$ is inaccessible, and indeed the existence of $\Sigma_n$-extendible cardinals is provable in ZFC via the reflection theorem. For example, every [[reflecting#Reflection and correctness | $\Sigma_n$ correct]] cardinal is $\Sigma_n$-extendible, since from $V_\kappa\prec_{\Sigma_n} V$ and $V_\lambda\prec_{\Sigma_n} V$, where $\kappa\lt\lambda$, it follows that $V_\kappa\prec_{\Sigma_n} V_\lambda$. So in fact there is a closed unbounded class of $\Sigma_n$-extendible cardinals. <br />
<br />
Similarly, every Mahlo cardinal $\kappa$ has a stationary set of inaccessible $\Sigma_n$-extendible cardinals $\gamma<\kappa$. <br />
<br />
$\Sigma_3$-extendible cardinals cannot be Laver [[indestructible]]. Therefore $\Sigma_3$-[[correct]], $\Sigma_3$-[[reflecting]], $0$-extendible, (pseudo-)[[uplifting]], [[weakly superstrong]], strongly uplifting, [[superstrong]], extendible, (almost) [[huge]] or [[rank-into-rank]] cardinals also cannot.<cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite><br />
<br />
=== $A$-extendible cardinals ===<br />
<br />
(this subsection from <cite>Hamkins2016:TheVopenkaPrincipleIs</cite> unless noted otherwise)<br />
<br />
Definitions:<br />
* A cardinal $κ$ is '''$A$-extendible''', for a class $A$, iff for every ordinal $λ > κ$ there is an ordinal $θ$ such that there is an elementary embedding<br />
*: $j : ⟨ V_λ , ∈, A ∩ V_λ ⟩ → ⟨ V_θ , ∈, A ∩ V_θ ⟩$<br />
*: with critical point $κ$ (such that $λ < j(κ)$ — ''removing this does not change, what cardinals are extendible'').<br />
** $λ$ is called the degree of $A$-extendibility of an embedding.<br />
* A cardinal $κ$ is '''$(Σ_n)$-extendible''', iff it is $A$-extendible, where $A$ is the $Σ_n$-truth predicate. (This is a different notion than $\Sigma_n$-extendible cardinals.)<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* For $A$-extendible $κ$ is, a set $g ⊆ κ$ is called '''$A$-stretchable''', if for every $λ > κ$ and every $h ⊆ λ$ for which $h ∩ κ = g$, there is an elementary embedding $j : ⟨V_λ , ∈, A ∩ V_λ⟩ → ⟨V_θ , ∈, A ∩ V_θ⟩$ such that $crit(j)=κ$, $λ < j(κ)$ and $j(g) ∩ λ = h$.<br />
** Intuitively, an $A$-stretchable set $g$ is one that can be stretched by an $A$-extendibility embedding to agree with any desired $h$ that extends $g$.<br />
** $A$-strechability is a form of [[Laver diamond]] for $A$-extendibility.<br />
* For $A$-extendible $κ$, a function $\ell : κ → V_κ$ is called an '''$A$-extendibility Laver function''', if for every $λ$ and every target $a$, there is an elementary embedding $j : ⟨V_λ , ∈, A ∩ V_λ⟩ → ⟨V_θ , ∈, A ∩ V_θ⟩$ such that $crit(j)=κ$, $λ < j(κ)$ and $j(\ell)(κ) = a$.<br />
<br />
Results:<br />
* The following notions are equivalent:<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
** $C^{(n)}$-extendibility in the sense of Bagaria (defined in a section above)<br />
** $A$-extendibility where $A$ is the class $C^{(n)}$<br />
** $(Σ_n)$-extendibility<br />
** $κ$ is $A$-extendible for every $Σ_n$-definable class $A$, allowing parameters in $V_κ$<br />
* The [[Vopenka|Vopěnka principle]] is equivalent over GBC to both following statements:<br />
** For every class $A$, there is an $A$-extendible cardinal.<br />
** For every class $A$, there is a stationary proper class of $A$-extendible cardinals.<br />
* If $κ$ is $A$-extendible for some class $A$, then<br />
** there is an $A$-stretchable set $g ⊆ κ$.<br />
** there is an $A$-extendibility Laver function $\ell : κ → V_κ$.<br />
* In $\text{GBC}$, for any class $A$ there is a class function $\ell : \mathrm{Ord} → V$, such that for every $A$-extendible cardinal $κ$, $\ell ↾ κ$ is an $A$-extendible Laver function for $κ$.<br />
** This uses global well-ordering that is a consequence of global choice.<br />
** Without global choice, one can still have ordinal-anticipating Laver function $\ell : \mathrm{Ord} → \mathrm{Ord}$ and get for example $A$-extendibility Menas property.<br />
* If $κ$ is $A$-extendible for a class $A$, then $κ$ is $Σ_2(A)$-[[reflecting]].<br />
* If $κ$ is $A ⊕ C$-extendible, where $C$ is the class of all $Σ_1(A)$-[[correct]] ordinals, then $κ$ is $Σ_3(A)$-reflecting.<br />
<br />
=== Virtually extendible cardinals ===<br />
<br />
Definitions:<br />
* A cardinal $κ$ is (weakly? strongly? ......) '''virtually extendible''' iff for every $α > κ$, in a set-forcing extension there is an elementary embedding $j : V_α → V_β$ with $\mathrm{crit(j)} = κ$ and $j(κ) > α$.<br />
** '''$C^{(n)}$-virtually extendible''' cardinals require additionally that $j(κ)$ has property $C^{(n)}$ (i.e. $\Sigma_n$-[[correct|correctness]]).<cite>GitmanSchindler:VirtualLargeCardinals</cite><br />
* A cardinal $κ$ is '''(weakly) virtually $A$-extendible''', for a class $A$, iff for every ordinal $λ > κ$ there is an ordinal $θ$ such that in a set-forcing extension, there is an elementary embedding<br />
*: $j : \langle V_λ , ∈, A ∩ V_λ \rangle → \langle V_θ , ∈, A ∩ V_θ \rangle$<br />
*: with critical point $κ$.<br />
** For '''(strongly) virtually $A$-extendible''' $κ$, we require additionally $λ < j(κ)$.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* A cardinal $κ$ is '''$n$-[[remarkable]]''', for $n > 0$, iff for every $η > κ$ in [[correct|$C^{(n)}$]] , there is $α<κ$ also in $C^{(n)}$ such that in $V^{Coll(ω, < κ)}$, there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$.<br />
** A cardinal is '''completely remarkable''' iff it is $n$-remarkable for all $n > 0$.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
* A cardinal κ is weakly or strongly virtually $(Σ_n)$-extendible, iff it is respectively weakly or strongly virtually $A$-extendible, where $A$ is the $Σ_n$-truth predicate.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
<br />
Equivalence and hierarchy:<br />
* $1$-remarkability is equivalent to remarkability. A cardinal is virtually $C^{(n)}$-extendible iff it is $n + 1$-remarkable (virtually extendible cardinals are virtually $C^{(1)}$-extendible).<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
* Weakly and strongly $A$-extendible cardinal are non-equivalent, although in the non-virtual context, the weak and strong forms of $A$-extendibility coincide.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* It is relatively consistent with GBC that every class $A$ admits a (weakly) virtually $A$-extendible cardinal (and so the generic Vopěnka principle holds), but no class $A$ admits a (strongly) virtually $A$-extendible cardinal.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* Every $n$-remarkable cardinal is in $C^{(n+1)}$.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
* Every $n+1$-remarkable cardinal is a limit of $n$-remarkable cardinals.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
<br />
Upper limits for strength:<br />
* If $κ$ is [[Shelah|virtually Shelah for supercompactness]] or 2-iterable, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$.<cite>GitmanSchindler:VirtualLargeCardinals</cite><br />
* If $κ$ is [[huge|virtually huge*]], then $V_κ$ is a model of proper class many virtually extendible cardinals.<cite>GitmanSchindler:VirtualLargeCardinals</cite><br />
* Completely remarkable cardinals can exist in $L$.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
* For a $2$-[[iterable]] cardinal $κ$, $V_κ$ is a model of proper class many completely remarkable cardinals.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
* If $0^\#$ exists, then every [[Silver indiscernible]] is in $L$ completely remarkable and virtually $A$-extendible for every definable class $A$.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple, GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
<br />
Lower limit for strength:<br />
* Virtually extendible cardinals are [[remarkable]] limits of remarkable cardinals and 1-[[iterable]] limits of 1-iterable cardinals.<cite>GitmanSchindler:VirtualLargeCardinals</cite><br />
<br />
In relation to [[Generic Vopěnka's Principle]]:(from <cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite> unless noted otherwise)<br />
* The following are equiconsistent<br />
** $gVP(Π_n)$<br />
** $gVP(κ, \mathbf{Σ_{n+1}})$ for some $κ$<br />
** There is an $n$-remarkable cardinal.<br />
* The following are equiconsistent<br />
** $gVP(\mathbf{Π_n})$<br />
** $gVP(κ, \mathbf{Σ_{n+1}})$ for a proper class of $κ$<br />
** There is a proper class of $n$-remarkable cardinals.<br />
* Unless there is a transitive model of ZFC with a proper class of $n$-remarkable cardinals,<br />
** if for some cardinal $κ$, $gVP(κ, \mathbf{Σ_{n+1}})$ holds, then there is an $n$-remarkable cardinal.<br />
** if $gVP(Π_n)$ holds, then there is an $n$-remarkable cardinal.<br />
** if $gVP(\mathbf{Π_n})$ holds, then there is a proper class of $n$-remarkable cardinals.<br />
* $κ$ is the least for which $gVP^∗(κ, \mathbf{Σ_{n+1}})$ holds. $\iff κ$ is the least $n$-remarkable cardinal.<br />
* If $gVP^∗(Π_n)$ holds, then there is an $n$-remarkable cardinal.<br />
* If $gVP^∗(\mathbf{Π_n})$ holds, then there is a proper class of $n$-remarkable cardinals.<br />
* If there is a proper class of $n$-remarkable cardinals, then $gVP(Σ_{n+1})$ holds.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* If $gVP(Σ_{n+1})$ holds, then either there is a proper class of $n$-remarkable cardinals or there is a proper class of [[rank-into-rank|virtually rank-into-rank]] cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* The generic Vopěnka principle holds iff for every class $A$, there are a proper class of (weakly) virtually $A$-extendible cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* The generic Vopěnka scheme is equivalent over ZFC to the scheme asserting of every definable class $A$ that there is a proper class of weakly virtually $A$-extendible cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* For $n ≥ 1$, the following are equivalent as schemes over ZFC:<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
** The generic Vopěnka scheme holds for $Π_{n+1}$-definable classes.<br />
** The generic Vopěnka scheme holds for $Σ_{n+2}$-definable classes.<br />
** For every $Σ_n$-definable class A, there is a proper class of (weakly) virtually $A$-extendible cardinals.<br />
** There is a proper class of (weakly) virtually $(Σ_n)$-extendible cardinals.<br />
** There is a proper class of cardinals $κ$, such that for every $Σ_n$-correct cardinal $λ>κ$, there is a $Σ_n$-correct cardinal $θ > λ$ and a virtual elementary embedding $j : V_λ → V_θ$ with $crit(j)=κ$.<br />
* If $0^♯$ exists, then there is a class-forcing extension $L[G]$ of the constructible universe in which the generic Vopěnka principle holds (so $gVP(κ, \mathbf{Σ_{n+1}})$ and $gVP(Π_n)$ hold for any $κ$ and $n$), but there are no $Σ_2$-reflecting cardinals and hence no remarkable cardinals (or $n$-remarkable cardinals).<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
<br />
== In set-theoretic geology ==<br />
If $κ$ is extendible then the $κ$-[[mantle]] of $V$ is its smallest ground (so of course the mantle is a ground of V).<cite>Usuba2018:ExtendibleCardinalsAndTheMantle</cite><br />
<br />
{{stub}}<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_cardinals&diff=4170Reflecting cardinals2022-05-29T13:27:58Z<p>BartekChom: /* $\Sigma_2$-correct cardinals */ indestructible</p>
<hr />
<div>{{DISPLAYTITLE: Reflecting cardinals}}<br />
[[Category:Middle attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting ordinal]]s.''<br />
Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak [[Kripke-Platek]] set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-separation after reflection down to a $V_\alpha$ containing the range of the defined function. Several philosophers have advanced philosophical justifications of large cardinals based on ideas arising from reflection.<br />
<br />
==Reflection theorem== <br />
The Reflection theorem is one of the most important theorems in Set Theory, being the basis for several large cardinals. The Reflection theorem is in fact a "meta-theorem," a theorem about proving theorems. The Reflection theorem intuitively encapsulates the idea that we can find sets resembling the class $V$ of all sets.<br />
<br />
'''Theorem (Reflection):''' For every set $M$ and formula $\phi(x_0...x_n,p)$ ($p$ is a parameter) there exists some limit ordinal $\alpha$ such that $V_\alpha\supseteq M$ such that $\phi^{V_\alpha}(x_0...x_n,p)\leftrightarrow \phi(x_0...x_n,p)$ (We say $V_\alpha$ reflects $\phi$). Assuming the Axiom of Choice, we can find some countable $M_0\supseteq M$ that reflects $\phi(x_0...x_n,p)$.<br />
<br />
Note that by conjunction, for any finite family of formulas $\phi_0...\phi_n$, as $V_\alpha$ reflects $\phi_0...\phi_n$ if and only if $V_\alpha$ reflects $\phi_0\land...\land\phi_n$. Another important fact is that the truth predicate for $\Sigma_n$ formulas is $\Sigma_{n+1}$, and so we can find a (Club class of) ordinals $\alpha$ such that $(V_\alpha,\in)\prec_{{T_{\Sigma_n}}\restriction{V_\alpha}} (V,\in)$, where $T_{\Sigma_n}$ is the truth predicate for $\Sigma_n$ and so $ZFC\vdash Con(ZFC(\Sigma_n))$ for every $n$, where $ZFC(\Sigma_n)$ is $ZFC$ with Replacement and Separation restricted to $\Sigma_n$.<br />
<br />
'''Lemma:''' If $W_\alpha$ is a cumulative hierarchy, there are arbitrarily large limit ordinals $\alpha$ such that $\phi^{W_\alpha}(x_0...x_n,p)\leftrightarrow \phi^W(x_0...x_n,p)$.<br />
<br />
==Reflection and correctness==<br />
<br />
For any class $\Gamma$ of formulas, an inaccessible cardinal $\kappa$ is ''$\Gamma$-reflecting'' if and only if $H_\kappa\prec_\Gamma V$, meaning that for any $\varphi\in\Gamma$ and $a\in H_\kappa$ we have $V\models\varphi[a]\iff H_\kappa\models\varphi[a]$. For example, an inaccessible cardinal is ''$\Sigma_n$-reflecting'' if and only if $H_\kappa\prec_{\Sigma_n} V$. In the case that $\kappa$ is not necessarily inaccessible, we say that $\kappa$ is ''$\Gamma$-correct'' if and only if $H_\kappa\prec_\Gamma V$''. <br />
<br />
* A simple L&ouml;wenheim-Skolem argument shows that every uncountable cardinal $\kappa$ is $\Sigma_1$-correct.<br />
* For each natural number $n$, the $\Sigma_n$-correct cardinals form a closed unbounded proper class of cardinals, as a consequence of the [[reflection theorem]]. This class is sometimes denoted by $C^{(n)}$ and the $\Sigma_n$-correct cardinals are also sometimes referred to as the $C^{(n)}$-cardinals. <br />
* Every $\Sigma_2$-correct cardinal is a [[beth fixed point | $\beth$-fixed point]] and a limit of such $\beth$-fixed points, as well as an [[aleph | $\aleph$-fixed point]] and a limit of such. Consequently, we may equivalently define for $n\geq 2$ that $\kappa$ is $\Sigma_n$-correct if and only if $V_\kappa\prec_{\Sigma_n} V$. <br />
<br />
A cardinal $\kappa$ is ''correct'', written $V_\kappa\prec V$, if it is $\Sigma_n$-correct for each $n$. This is not expressible by a single assertion in the language of set theory (since if it were, the least such $\kappa$ would have to have a smaller one inside $V_\kappa$ by elementarity). Nevertheless, $V_\kappa\prec V$ is expressible as a scheme in the language of set theory with a parameter (or constant symbol) for $\kappa$. <br />
<br />
Although it may be surprising, the existence of a correct cardinal is equiconsistent with ZFC. This can be seen by a simple compactness argument, using the fact that the theory ZFC+"$\kappa$ is correct" is finitely consistent, if ZFC is consistent, precisely by the observation about $\Sigma_n$-correct cardinals above.<br />
<br />
[[C^(n)|$C^{(n)}$]] are the classes of $\Sigma_n$-correct ordinals. These classes are clubs (closed unbounded). $C^{(0)}$ is the class of all ordinals. $C^{(1)}$ is precisely the class of all uncountable cardinals $α$ such that $V_\alpha=H(\alpha)$; i.e. precisely the Beth fixed points. References to the $C^{(n)}$ classes (different from just the requirement that the cardinal belongs to $C^{(n)}$) can sometimes make large cardinal properties stronger (for example $C^{(n)}$-[[superstrong]], $C^{(n)}$-[[supercompact]], $C^{(n)}$-[[extendible]], $C^{(n)}$-[[huge]] and $C^{(n)}$-[[rank-into-rank]] cardinals). On the other hand, every [[measurable]] cardinal is $C^{(n)}$-measurable for all $n$ and every ($λ$-)[[strong]] cardinal is ($λ$-)$C^{(n)}$-strong for all $n$.<cite>Bagaria2012:CnCardinals</cite><br />
<br />
A cardinal $\kappa$ is ''reflecting'' if it is inaccessible and correct. Just as with the notion of correctness, this is not first-order expressible as a single assertion in the language of set theory, but it is expressible as a scheme (''Lévy scheme''). The existence of such a cardinal is equiconsistent to the assertion [[ORD is Mahlo]].<br />
<br />
If there is a pseudo [[uplifting]] cardinal, or indeed, merely a pseudo $0$-uplifting cardinal $\kappa$, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. You can get this by taking some $\lambda\gt\kappa$ such that $V_\kappa\prec V_\lambda$.<br />
<br />
== $\Sigma_2$-correct cardinals == <br />
<br />
The $\Sigma_2$-correct cardinals are a particularly useful and robust class of cardinals, because of the following characterization: $\kappa$ is $\Sigma_2$-correct if and only if for any $x\in V_\kappa$ and any formula $\varphi$ of any complexity, whenever there is an ordinal $\alpha$ such that $V_\alpha\models\varphi[x]$, then there is $\alpha\lt\kappa$ with $V_\alpha\models\varphi[x]$. The reason this is equivalent to $\Sigma_2$-correctness is that assertions of the form $\exists \alpha\ V_\alpha\models\varphi(x)$ have complexity $\Sigma_2(x)$, and conversely all $\Sigma_2(x)$ assertions can be made in that form. <br />
<br />
It follows, for example, that if $\kappa$ is $\Sigma_2$-correct, then any feature of $\kappa$ or any larger cardinal than $\kappa$ that can be verified in a large $V_\alpha$ will reflect below $\kappa$. So if $\kappa$ is $\Sigma_2$-reflecting, for example, then there must be unboundedly many inaccessible cardinals below $\kappa$. Similarly, if $\kappa$ is $\Sigma_2$-reflecting and measurable, then there must be unboundedly many measurable cardinals below $\kappa$.<br />
<br />
One can also say that a $Σ_2$-reflecting cardinal is a regular cardinal $κ$ such that $V_κ \preccurlyeq_{Σ_2} V$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
Other facts:<br />
* [[Remarkable]] cardinals are $Σ_2$-reflecting.<cite>Wilson2018:WeaklyRemarkableCardinals</cite><br />
* It is relatively consistent that ZFC and the [[Vopenka|generic Vopěnka scheme]] holds, yet [[Ord is Mahlo|$Ord$ is not definably Mahlo]] and not even $∆_2$-Mahlo. In such a model, there can be no $Σ_2$-reflecting cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* An [[Axioms of generic absoluteness|axiom of generic absoluteness]], $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$, is equiconsistent with the existence of a $Σ_2$-reflecting cardinal.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
As for $\Sigma_3$-correctness, $\Sigma_3$-correct cardinals (among others) cannot be Laver [[indestructible]], because $\Sigma_3$-[[extendible]] cardinals cannot.<cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite><br />
<br />
==The Feferman theory==<br />
<br />
This is the theory, expressed in the language of set theory augmented with a new unary class predicate symbol $C$, asserting that $C$ is a closed unbounded class of cardinals, and every $\gamma\in C$ has $V_\gamma\prec V$. In other words, the theory consists of the following scheme of assertions: $$\forall\gamma\in C\ \forall x\in V_\gamma\ \bigl[\varphi(x)\iff\varphi^{V_\gamma}(x)\bigr]$$<br />
as $\varphi$ ranges over all formulas. Thus, the Feferman theory asserts that the universe $V$ is the union of a chain of elementary substructures $$V_{\gamma_0}\prec V_{\gamma_1}\prec\cdots\prec V_{\gamma_\alpha}\prec\cdots \prec V$$<br />
Although this may appear at first to be a rather strong theory, since it seems to imply at the very least that each $V_\gamma$ for $\gamma\in C$ is a model of ZFC, this conclusion would be incorrect. In fact, the theory does ''not'' imply that any $V_\gamma$ is a model of ZFC, and does not prove $\text{Con}(\text{ZFC})$; rather, the theory implies for each axiom of ZFC separately that each $V_\gamma$ for $\gamma\in C$ satisfies it. Since the theory is a scheme, there is no way to prove from that theory that any particular $\gamma\in C$ has $V_\gamma$ satisfying more than finitely many axioms of ZFC. In particular, a simple compactness argument shows that the Feferman theory is consistent provided only that ZFC itself is consistent, since any finite subtheory of the Feferman theory is true by the [[reflection theorem]] in any model of ZFC. It follows that the Feferman theory is actually conservative over ZFC, and proves with ZFC no new facts about sets that is not already provable in ZFC alone. <br />
<br />
The Feferman theory was proposed as a natural theory in which to undertake the category-theoretic uses of [[Grothendieck universe | Grothendieck universes]], but without the large cardinal penalty of a proper class of inaccessible cardinals. Indeed, the Feferman theory offers the advantage that the universes are each elementary substructures of one another, which is a feature not generally true under the [[universe axiom]].<br />
<br />
==Maximality Principle==<br />
<br />
The existence of an inaccessible reflecting cardinal is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if $\kappa$ is an inaccessible reflecting cardinal, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model with an inaccessible reflecting cardinal.<br />
<br />
== $Σ_n(A)$-correct ==<br />
(this section from <cite>Hamkins2016:TheVopenkaPrincipleIs</cite>)<br />
<br />
Definitions:<br />
* An ordinal $γ$ is $Σ_n(A)$-correct, if $⟨V_γ, ∈, A ∩ V_γ⟩ ≺_{Σ_n} ⟨V, ∈, A⟩$.<br />
* A cardinal $κ$ is $Σ_n(A)$-reflecting, if it is inaccessible and $Σ_n(A)$-correct.<br />
<br />
Results:<br />
* If $κ$ is $A$-[[extendible]] for a class $A$, then $κ$ is $Σ_2(A)$-reflecting.<br />
* If $κ$ is $A ⊕ C$-extendible, where $C$ is the class of all $Σ_1(A)$-correct ordinals, then $κ$ is $Σ_3(A)$-reflecting.<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Weakly_measurable&diff=4169Weakly measurable2022-05-29T13:27:45Z<p>BartekChom: /* Weakly measurable cardinals and forcing */ indestructible</p>
<hr />
<div>{{DISPLAYTITLE: Weakly measurable cardinals}}<br />
<br />
The weakly measurable cardinals were introduced by Jason Schanker in <CITE>Schanker2011:WeaklyMeasurableCardinals</CITE>, <CITE>Schanker2011:Thesis</CITE>. As their name suggests, they provide a weakening of the large cardinal concept of [[measurable|measurability]]. If the GCH holds at $\kappa$, then the property of the weak measurability of $\kappa$ is equivalent to that of the full measurability of $\kappa$, but when $\kappa^+\lt 2^\kappa$, these concepts can separate. Nevertheless, the existence of a weakly measurable cardinal is equiconsistent with the existence of a measurable cardinal, since if $\kappa$ is weakly measurable, then it is measurable in an inner model.<br />
<br />
== Formal Definition ==<br />
<br />
A cardinal $\kappa$ is ''weakly measurable'' if and only if for every family $\mathcal{A}\subset P(\kappa)$ of size at most $\kappa^+$, there is a nonprincipal $\kappa$-complete [[filter]] on $\kappa$ measuring every set in $\mathcal{A}$. (i.e., For every subset $A \in \mathcal{A}$, either $A$ or $\kappa \setminus A$ is in the filter.)<br />
<br />
== Embedding characterizations of weak measurability==<br />
<br />
If $(\kappa^+)^{{<}\kappa} = \kappa^+$, then weak measurability can also be equivalently characterized in several different ways in terms of [[elementary embedding|elementary embeddings]]. <br />
<br />
:; Weak embedding characterization : For every $A \subseteq \kappa^+$, there exists a transitive $M \vDash \text{ZFC}^-$ with $A, \kappa \in M$, a transitive $N$ and an [[elementary embedding]] $j: M \longrightarrow N$ with critical point $\kappa$.<br />
<br />
:; Embedding characterization : For every transitive set $M$ of size $\kappa^+$ with $\kappa \in M$, there exists a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.<br />
<br />
:; Normal embedding characterization : For every transitive $M \vDash \text{ZFC}^-$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $\kappa \in M$, there exists a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.<br />
<br />
:; Normal ZFC embedding characterization : For every $A \subseteq H_{\kappa^+}$ of size $\kappa^+$, there exists a transitive $M \vDash \text{ZFC}$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $A \subseteq M$ and $\kappa \in M$, a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences, and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.<br />
<br />
== Weakly measurable cardinals and inner models ==<br />
<br />
Weakly measurable cardinals are incompatible with the axiom [[V = L|$V = L$]] since such cardinals are fully measurable if the GCH holds, and the constructible universe cannot contain nonprincipal countably complete ultrafilters. By the same reasoning, the Dodd-Jensen core model $K^{DJ}$ will not have any cardinals that it thinks are weakly measurable. If $\kappa$ is weakly measurable, then we can always find a countably complete normal $K^{DJ}$-ultrafilter $U$ whereby $\kappa$ will be measurable in $L[U]$ (<CITE>Mitchell2001:TheCoveringLemma</CITE>, Lemma 3.36). Under certain anti-large cardinal hypotheses, a weakly measurable cardinal will be measurable in the suitable core model. For example, if $\kappa$ is weakly measurable and there is no inner model with a measurable cardinal $\lambda$ having [[Mitchell_order|Mitchell order]] $\lambda^{++}$, then $\kappa$ will be measurable in Mitchell's core model $K^m$ (<CITE>Jech2003:SetTheory</CITE>, Theorem 35.17).<br />
<br />
== Weakly measurable cardinals and forcing ==<br />
<br />
Weakly measurable cardinals $\kappa$ are invariant under forcing of size less than $\kappa$ and forcing that adds no new subsets of $\kappa^+$. Many other preservation results for these large cardinals are unknown. For example, it is an open question as to whether we can always force to an extension where a weakly measurable cardinal $\kappa$ from the ground model remains weakly measurable and becomes [[indestructible]] by the further forcing to add a Cohen subset of $\kappa$. However, if $\kappa$ is measurable in the ground model, we inherit all of the indestructibility results we can get for its weak measurability from its full measurability and more. In particular, we will be able to force to an extension where $\kappa$ is measurable, the GCH holds, and the '''weak''' measurability of $\kappa$ is preserved by the further forcing to add any number of Cohen subsets of $\kappa$. Starting with a measurable cardinal $\kappa$, this result allows us to force to an extension where we preserve the weak measurability of $\kappa$ and yet make the GCH fail first at $\kappa$. Since the GCH cannot fail first at a measurable cardinal, this will also be a forcing extension where $\kappa$ is no longer measurable.<br />
<br />
== Place in the large cardinal hierarchy ==<br />
<br />
In terms of consistency strength, weakly measurable cardinals occupy the same place as measurable cardinals in the large cardinal hierarchy. In terms of size, the possibilities for these large cardinals are still being investigated. Because measurable cardinals must be weakly measurable, and weakly measurable cardinals must be weakly compact, we are provided with strict upper and lower bounds on their sizes with respect to these large cardinal notions. In the presence of the GCH, weakly measurable cardinals and measurable cardinals coincide so their sizes are the same in this case. At the opposite extreme, it was left as an open question in <CITE>Schanker2011:WeaklyMeasurableCardinals</CITE> and <CITE>Schanker2011:Thesis</CITE> as to whether the least weakly measurable cardinal could also be the least weakly compact cardinal. Despite being left open, there are promising developments that are being undertaken jointly by Gitik, Hamkins, and Schanker, which are aimed at this possibility.<br />
<br />
{{References}}<br />
<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Unfoldable&diff=4168Unfoldable2022-05-29T13:26:53Z<p>BartekChom: /* Relation to forcing */ indestructible</p>
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<div>{{DISPLAYTITLE: Unfoldable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
The unfoldable cardinals were introduced by Andres Villaveces in order to generalize the definition of [[weakly compact|weak compactness]]. Because weak compactness has many different definitions, the one he chose to extend was specifically the embedding property &#40;see weakly compact for more information). The way he did this was analogous to the generalization of [[huge]] cardinals to superhuge cardinals.<br />
<br />
== Definition ==<br />
<br />
There are unfoldable cardinals and strongly unfoldable cardinals, as well as superstrongly unfoldable &#40;AKA almost-hugely unfoldable AKA [[uplifting|strongly uplifting]]) cardinals. All of these are generalizations of [[weakly compact|weak compactness]].<br />
<br />
=== Unfoldable ===<br />
<br />
A cardinal $\kappa$ is '''$\theta$-unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j&#40;\kappa)\geq\theta$. $\kappa$ is then called '''unfoldable''' iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.<br />
<br />
Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=&#40;M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $&#40;\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=&#40;N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:<br />
<br />
*$\mathcal{M}$ is an elementary substructure of $\mathcal{N}$<br />
*$\mathcal{M}\neq\mathcal{N}$<br />
*For any $a\in M$, $b\in N$, and $\gamma<\beta$, $b R_\gamma^\mathcal{N} a\rightarrow b\in M$<br />
<br />
If such holds, $\mathcal{M}$ is said to be '''nontrivially end elementary extended''' by $\mathcal{N}$ and '''$\mathcal{N}$ is a nontrivial end elementary extension''' of $\mathcal{M}$, abbreviated $\mathcal{N}$ is an '''eee''' of $\mathcal{M}$.<br />
<br />
A cardinal $\kappa$ is '''$\lambda$-unfoldable''' iff $\kappa$ is [[inaccessible]] and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $&#40;V_\kappa;\in,S)$ such that $M\not\in V_\lambda$. $\kappa $ is '''unfoldable''' iff $M $ can be made to have arbitrarily large rank. In this case, $&#40;V_\kappa;\in,S)\prec_e &#40;M;\in^M,S^M)$ iff $&#40;V_\kappa;\in,S)\prec &#40;M;\in^M,S^M)$ and $&#40;V_\kappa;\in)\prec_e &#40;M;\in^M)$. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
$\kappa$ is also '''unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $&#40;V_\kappa;\in,S)$, $&#40;\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $&#40;V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
=== Long Unfoldable ===<br />
<br />
$\kappa$ is '''long unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $&#40;V_\kappa;\in,S)$, $&#40;\mathcal{E};\prec_e)$ has chains of length $\text{Ord}$. <br />
<br />
Every long unfoldable cardinal is unfoldable. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
=== Strongly Unfoldable ===<br />
<br />
A cardinal $\kappa$ is '''$\theta$-strongly unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j&#40;\kappa)\geq\theta$ and $V_\theta\subseteq N$.<br />
<br />
$\kappa$ is then called '''strongly unfoldable''' iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.<br />
<br />
As defined in <cite>HamkinsJohnstone2010:IndestructibleStrongUnfoldability</cite> in analogy with [[Mitchell rank]]s, a strongly unfoldable cardinal $\kappa$ is '''strongly unfoldable of degree $\alpha$''', for an ordinal $\alpha$, if for every ordinal $\theta$ it is $\theta$-strongly unfoldable of degree $\alpha$, meaning that for each $A \in H_{\kappa^+}$ there is a $\kappa$-[[model]] $M \models \mathrm{ZFC}$ with $A \in M$ and a transitive set $N$ with $\alpha \in M$ and an elementary embedding $j:M \to N$ having critical point $\kappa$ with $j&#40;\kappa)>\max\{\theta, \alpha\}$ and $V_\theta \subset N$, such that $\kappa$ is strongly unfoldable of every degree $\beta < \alpha$ in $N$.<cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite><br />
<br />
The strongly unfoldable cardinals are exactly the [[shrewd]] cardinals. [https://arxiv.org/abs/2107.12722]<br />
<br />
=== Superstrongly Unfoldable ===<br />
<br />
Superstrongly unfoldable and almost-hugely unfoldable cardinals are defined and shown to be equivalent to [[uplifting|strongly uplifting]] &#40;described there) in <cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite>.<br />
<br />
== Relations to Other Cardinals ==<br />
Here is a list of relations between unfoldability and other large cardinal axioms:<br />
<br />
*For every finite $m$ and $n$, unfoldability implies the consistency of the existence of a [[indescribable|$\Pi_m^n$-indescribable]] cardinal &#40;specifically, such cardinals exist in $V_\kappa\cap L$ for some $\kappa$). Furthermore, if $V=L$, then the least $\Pi_m^n$-indescribable cardinal is less than the least unfoldable cardinal, and every unfoldable cardinal is totally indescribable. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Any strongly unfoldable cardinal is [[indescribable|totally indescribable]] and a limit of totally indescribable cardinals. Therefore the consistency strength of unfoldability is stronger than total indescribability. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Every superstrongly unfoldable cardinal &#40;i.e. [[uplifting|strongly uplifting]] cardinal) is strongly unfoldable of every ordinal degree <math>\alpha</math>, and a stationary limit of cardinals that are strongly unfoldable of every ordinal degree and so on. <cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite><br />
*The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable, which are precisely those that are [[weakly compact]]. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. <cite>Hamkins2008:UnfoldableGCH</cite><br />
*The assertion that a [[Ramsey]] cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Although unfoldable cardinals are consistency-wise stronger than [[weakly compact]] cardinals, if there is a strongly unfoldable cardinal, then in a forcing extension, the least weakly compact cardinal is also the least unfoldable cardinal.<cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*The existence of a [[subtle]] cardinal is consistency-wise stronger than the existence of an unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*If a [[subtle]] cardinal and an unfoldable cardinal exist and $V=L$, then the least unfoldable cardinal is larger than the least subtle cardinal &#40;and therefore much larger than the least [[weakly compact]]). <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Any [[Ramsey]] cardinal is unfoldable. If there is a weakly compact cardinal above an [[Erdos|$\omega_1$-Erdos]] cardinal, then the least unfoldable is less than that &#40;therefore less than the least Ramsey). <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Even though it may seem odd at first, if both exist then the least [[Rank-into-rank|I3]] cardinal is less than the least strongly unfoldable cardinal.<br />
* $ω_1$-[[iterable]] cardinals are strongly unfoldable in $L$.<cite>GitmanWelch2011:RamseyLikeCardinalsII</cite><br />
<br />
== Relation to forcing ==<br />
From <cite>Hamkins2008:UnfoldableGCH</cite>:<br />
* If $κ$ is unfoldable, then in a forcing extension by forcing of size $κ$ the unfoldability of $κ$ is [[indestructible]] by $\mathrm{Add}&#40;κ, θ)$ for any $θ$. &#40;Main Theorem 2)<br />
* If $κ$ is strongly unfoldable, then in a forcing extension by forcing of size $κ$ the strong unfoldability of $κ$ is indestructible by $\mathrm{Add}&#40;κ, 1)$ but not by $\mathrm{Add}&#40;κ, θ)$ for any cardinal $θ > κ$. &#40;Theorem 7)<br />
* Any unfoldable cardinal can be made, by forcing of size $κ^+$, $κ$ unfoldable but not strongly unfoldable. &#40;Corollary 8)<br />
* In a forcing extension preserving all strongly unfoldable cardinals, GCH fails at every [[inaccessible]] cardinal. &#40;Theorem 9)<br />
* Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every inaccessible cardinal. Therefore, GCH can fail at every unfoldable cardinal.<br />
<br />
''TODO: connection to weak forms of PFA''<br />
<br />
''TODO: consistency with slim Kurepa trees''<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Inner_model&diff=4167Inner model2022-05-29T13:24:05Z<p>BartekChom: Redirected page to Model#Class-sized transitive models</p>
<hr />
<div>#REDIRECT [[Model#Class-sized_transitive_models]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Model&diff=4166Model2022-05-29T13:22:35Z<p>BartekChom: /* Class-sized transitive models */ indestructible supercompact ±GCH</p>
<hr />
<div>A '''model''' of a theory $T$ is a set $M$ together with relations (eg. two: $a$ and $b$) satisfying all axioms of the theory $T$. Symbolically $\langle M, a, b \rangle \models T$. According to the Gödel completeness theorem, in $\mathrm{PA}$ ([[Peano arithmetic]]; also in theories containing $\mathrm{PA}$, like $\mathrm{ZFC}$) a theory has models iff it is consistent. According to Löwenheim–Skolem theorem, in $\mathrm{ZFC}$ if a countable first-order theory has an infinite model, it has infinite models of all cardinalities.<br />
<br />
A '''model''' of a set theory (eg. $\mathrm{ZFC}$) is a set $M$ such that the structure $\langle M,\hat\in \rangle$ satisfies all axioms of the set theory. If $\hat \in$ is base theory's $\in$, the model is called a '''transitive model'''. Gödel completeness theorem and Löwenheim–Skolem theorem do not apply to transitive models. (But Löwenheim–Skolem theorem together with Mostowski collapsing lemma show that if there is a transitive model of ZFC, then there is a countable such model.) See [[Transitive ZFC model]] and [[Heights of models]].<br />
<br />
== Class-sized transitive models ==<br />
One can also talk about class-sized transitive models. Inner model is a [[transitive]] class (from other point of view, a class-sized transitive model (of ZFC or a weaker theory)) containing all ordinals. [[Forcing]] creates outer models, but it can also be used in relation with inner models.<cite>FuchsHamkinsReitz2015:SetTheoreticGeology</cite><br />
<br />
Among them are ''canonical inner models'' like<br />
* the [[core model]]<br />
* the canonical model [[constructible universe|$L[\mu]$]] of one measurable cardinal <br />
* [[HOD]] and generic HOD (gHOD)<br />
* mantle $\mathbb{M}$ (=generic mantle $g\mathbb{M}$)<br />
* outer core<br />
* the [[constructible universe]] $L$<br />
<br />
Some properties usually obtained by forcing are possible in inner models, for example<cite>ApterGitmanHamkins2012:InnerModelsUsuallyForcing</cite>:<br />
* (theorem 14) If there is a [[supercompact]] cardinal, then there are inner models with an [[indestructible]] supercompact cardinal $κ$ such that<br />
** $2^κ = κ^+$<br />
** $2^κ = κ^{++}$<br />
** Moreover, for every cardinal $θ$, such inner models $W$ can be found for which also $W^θ ⊆ W$.<br />
<br />
=== Mantle ===<br />
The '''mantle''' $\mathbb{M}$ is the intersection of all grounds. Mantle is always a model of ZFC. Mantle is a ground (and is called a '''bedrock''') iff $V$ has only set many grounds.<cite>FuchsHamkinsReitz2015:SetTheoreticGeology, Usuba2017:DDGandVeryLarge</cite><br />
<br />
'''Generic mantle''' $g\mathbb{M}$ was defined as the intersection of all mantles of generic extensions, but then it turned out that it is identical to the mantle.<cite>FuchsHamkinsReitz2015:SetTheoreticGeology, Usuba2017:DDGandVeryLarge</cite><br />
<br />
'''$α$th inner mantle''' $\mathbb{M}^α$ is defined by $\mathbb{M}^0=V$, $\mathbb{M}^{α+1} = \mathbb{M}^{\mathbb{M}^α}$ (mantle of the previous inner mantle) and $\mathbb{M}^α = \bigcap_{β<α} \mathbb{M}^β$ for limit $α$. If there is uniform presentation of $\mathbb{M}^α$ for all ordinals $α$ as a single class, one can talk about $\mathbb{M}^\mathrm{Ord}$, $\mathbb{M}^{\mathrm{Ord}+1}$ etc. If an inner mantle is a ground, it is called the '''outer core'''.<cite>FuchsHamkinsReitz2015:SetTheoreticGeology</cite><br />
<br />
It is conjenctured (unproved) that every model of ZFC is the $\mathbb{M}^α$ of another model of ZFC for any desired $α ≤ \mathrm{Ord}$, in which the sequence of inner mantles does not stabilise before $α$. It is probable that in the some time there are models of ZFC, for which inner mantle is undefined (Analogously, a 1974 result of Harrington appearing in (Zadrożny, 1983, section 7), with related work in (McAloon, 1974), shows that it is relatively consistent with Gödel-Bernays set theory that $\mathrm{HOD}^n$ exists for each $n < ω$ but the intersection $\mathrm{HOD}^ω = \bigcap_n \mathrm{HOD}^n$ is not a class.).<cite>FuchsHamkinsReitz2015:SetTheoreticGeology</cite><br />
<br />
For a cardinal $κ$, we call a ground $W$ of $V$ a $κ$-ground if there is a poset $\mathbb{P} ∈ W$ of size $< κ$ and a $(W, \mathbb{P})$-generic $G$ such that $V = W[G]$. The '''$κ$-mantle''' is the intersection of all $κ$-grounds.<cite>Usuba2018:ExtendibleCardinalsAndTheMantle</cite><br />
<br />
The $κ$-mantle is a definable, transitive, and extensional class. It is consistent that the $κ$-mantle is a model of ZFC (e.g. when there are no grounds), and if $κ$ is a strong limit, then the $κ$-mantle must be a model of ZF. However it is not known whether the $κ$-mantle is always a model of ZFC.<cite>Usuba2018:ExtendibleCardinalsAndTheMantle</cite><br />
<br />
==== Mantle and large cardinals ====<br />
If $\kappa$ is [[hyperhuge]], then $V$ has $<\kappa$ many [[ground]]s (so the mantle is a ground itself).<cite>Usuba2017:DDGandVeryLarge</cite><br />
<br />
If $κ$ is [[extendible]] then the $κ$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of $V$).<cite>Usuba2018:ExtendibleCardinalsAndTheMantle</cite><br />
<br />
On the other hand, it s consistent that there is a [[supercompact]] cardinal and class many grounds of $V$ (because of the indestructibility properties of supercompactness).<cite>Usuba2017:DDGandVeryLarge</cite><br />
<br />
== $\kappa$-model ==<br />
A '''weak $κ$-model''' is a [[transitive]] set $M$ of size $\kappa$ with $\kappa \in M$ and satisfying the theory $\mathrm{ZFC}^-$ ($\mathrm{ZFC}$ without the axiom of power set, with collection, not replacement). It is a '''$κ$-model''' if additionaly $M^{<\kappa} \subseteq M$.<cite>HamkinsJohnstone:BoldfaceResurrectionAxioms, HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
== Prime models and minimal models ==<br />
(from <cite>Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals</cite>, p. 23-24 unless noted otherwise)<br />
<br />
A '''minimal model''' is one without proper elementary submodels.<br />
<br />
A '''prime model''' is one that [[elementary embedding|embeds elementarily]] into every model of its theory. (compare <cite>HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory</cite>, p. 4)<br />
<br />
In general:<br />
* First order theories need not have either prime or minimal models.<br />
* Prime models need not be minimal, and minimal models need not be prime.<br />
<br />
However, for a model $\mathfrak{M} \models \text{ZF}$, $\mathfrak{M}$ is a prime model $\implies$ $\mathfrak{M}$ is a [[Paris model]] and satisfies AC $\implies$ $\mathfrak{M}$ is a minimal model.<br />
* Neither implication reverses in general, but both do if $\mathfrak{M} \models V=HOD$.<br />
<br />
The [[minimal transitive model of ZFC]] is an important model.<br />
<br />
== Solovay model ==<br />
&#40;from <cite>BagariaBosch2004:PFESolovay</cite>)<br />
<br />
......<br />
<br />
<!-- p. 2 -->Deﬁnition: $L(\mathbb{R})^M$ is a ''Solovay model'' over $V$ for $V⊆M$ and $M$ satisfying: $\forall_{x∈\mathbb{R}}$ $ω_1$ is an inaccessible cardinal in $V[x]$ and $x$ is ''small-generic'' over $V$ &#40;<!-- p. 1 -->there is a forcing notion $\mathbb{P}$ in $V$ countable in $M$ and there is, in $M$, a $\mathbb{P}$-generic ﬁlter $g$ over $V$ such that $x∈V[g]$).<br />
<br />
......<br />
<br />
{{References}}<br />
<br />
[[Category:The cellar]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Library&diff=4165Library2022-05-29T13:05:22Z<p>BartekChom: /* Library holdings */ Inner models with large cardinal features usually obtained by forcing</p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}[[Category:Cantor's Attic]]<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
Welcome to the library, our central repository for references cited here on Cantor's attic.<br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
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<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0460120 &#40;57 \#116)},<br />
MRREVIEWER = {Thomas J. Jech}<br />
}<br />
<br />
#Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod bibtex=@article {Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod,<br />
AUTHOR = {Apter, Arthur W.},<br />
TITLE = {Some applications of Sargsyan’s equiconsistency method},<br />
JOURNAL = {Fund. Math.},<br />
VOLUME = {216},<br />
PAGES = {207--222},<br />
}<br />
<br />
#ApterGitmanHamkins2012:InnerModelsUsuallyForcing bibtex=@ARTICLE{ApterGitmanHamkins2012:InnerModelsUsuallyForcing,<br />
title = "Inner models with large cardinal features usually obtained by forcing",<br />
author = "Apter, Arthur W and Gitman, Victoria and Hamkins, Joel David",<br />
journal = "Arch. Math. Logic",<br />
publisher = "Springer Science and Business Media LLC",<br />
volume = "51",<br />
number = "3-4",<br />
pages = "257--283",<br />
month = "may",<br />
year = "2012",<br />
language = "en",<br />
eprint = "1111.0856"<br />
}<br />
<br />
#Arai97:P bibtex=@paper{Arai97:P,<br />
title={A sneak preview of proof theory of ordinals},<br />
author={Arai, Toshiyasu},<br />
url={https://www.arxiv.org/abs/1102.0596v1},<br />
year={1997}<br />
}<br />
<br />
#Arai2019:FirstOrderReflection bibtex=@paper{Arai2019:FirstOrderReflection,<br />
TITLE = {A simplified ordinal analysis of first-order reflection},<br />
AUTHOR = {Arai, Toshiyasu},<br />
URL = {https://arxiv.org/abs/1907.07611v1},<br />
YEAR = {2019}<br />
}<br />
<br />
#Baaz2011:Kurt bibtex=@book{Baaz2011:Kurt,<br />
title={Kurt Gödel and the Foundations of Mathematics: Horizons of Truth},<br />
author={Baaz, M. and Papadimitriou, C.H. and Putnam, H.W. and Scott, D.S. and Harper, C.L.},<br />
isbn={9781139498432},<br />
url={https://books.google.pl/books?id=Tg0WXU5\_8EgC},<br />
year={2011},<br />
publisher={Cambridge University Press}<br />
}<br />
<br />
#Bagaria2002:AxiomsOfGenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {Axioms of generic absoluteness},<br />
JOURNAL = {Logic Colloquium 2002},<br />
BOOKTITLE = {Logic Colloquium '02: Lecture Notes in Logic 27},<br />
YEAR = {2006},<br />
DOI = {10.1201/9781439865903},<br />
ISBN = {9780429065262},<br />
URL = {https://www.academia.edu/2561575/AXIOMS_OF_GENERIC_ABSOLUTENESS},<br />
}<br />
<br />
#BagariaBosch2004:PFESolovay bibtex=@article {BagariaBosch2004:PFESolovay,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Proper forcing extensions and Solovay models},<br />
JOURNAL = {Archive for Mathematical Logic},<br />
YEAR = {2004},<br />
DOI = {10.1007/s00153-003-0210-2},<br />
URL = {https://www.academia.edu/2561570/Proper_forcing_extensions_and_Solovay_models},<br />
}<br />
<br />
#BagariaBosch2007:GenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Generic absoluteness under projective forcing},<br />
JOURNAL = {Fundamenta Mathematicae},<br />
YEAR = {2007},<br />
VOLUME = {194},<br />
PAGES = {95-120},<br />
DOI = {10.4064/fm194-2-1},<br />
}<br />
<br />
#Bagaria2012:CnCardinals bibtex=@article{Bagaria2012:CnCardinals,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {$C^{&#40;n)}$-cardinals},<br />
journal = {Archive for Mathematical Logic},<br />
YEAR = {2012},<br />
volume = {51},<br />
number = {3--4},<br />
pages = {213--240},<br />
DOI = {10.1007/s00153-011-0261-8},<br />
URL = {http://www.mittag-leffler.se/sites/default/files/IML-0910f-26.pdf},<br />
eprint = {1908.09664}<br />
}<br />
<br />
#BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@article{BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br />
AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosický, Jiří},<br />
TITLE = {Definable orthogonality classes in accessible categories are small},<br />
journal = {Journal of the European Mathematical Society},<br />
volume = {17},<br />
number = {3},<br />
pages = {549--589},<br />
eprint = {1101.2792}<br />
}<br />
<br />
#BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible bibtex=@article{BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible,<br />
author = {Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi},<br />
title = {Superstrong and other large cardinals are never Laver indestructible},<br />
eprint = {1307.3486},<br />
year = {2013},<br />
journal = {Archive for Mathematical Logic},<br />
volume = {55},<br />
number = {1-2},<br />
pages = {19--35},<br />
url = {http://jdh.hamkins.org/superstrong-never-indestructible/},<br />
doi = {10.1007/s00153-015-0458-3}<br />
}<br />
<br />
#Bagaria2017:LargeCardinalsBeyondChoice bibtex=@article{Bagaria2017:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2017},<br />
url = {https://events.math.unipd.it/aila2017/sites/default/files/BAGARIA.pdf}<br />
}<br />
<br />
#BagariaGitmanSchindler2017:VopenkaPrinciple bibtex=@ARTICLE{BagariaGitmanSchindler2017:VopenkaPrinciple,<br />
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},<br />
TITLE = {Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
NUMBER = {1-2},<br />
PAGES = {1--20},<br />
ISSN = {0933-5846},<br />
MRCLASS = {03E35 &#40;03E55 03E57)},<br />
MRNUMBER = {3598793},<br />
DOI = {10.1007/s00153-016-0511-x},<br />
URL = {https://victoriagitman.github.io/publications/2016/02/10/generic-vopenkas-principle-remarkable-cardinals-and-the-weak-proper-forcing-axiom.html}<br />
}<br />
<br />
#BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice bibtex=@article{BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan and Koellner, Peter and Woodin, W. Hugh},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2019},<br />
journal = {Bulletin of Symbolic Logic},<br />
volume = {25},<br />
number = {3},<br />
pages = {283--318},<br />
url = {https://par.nsf.gov/servlets/purl/10149501},<br />
doi = {10.1017/bsl.2019.28}<br />
}<br />
<br />
#Baumgartner1975:Ineffability bibtex=@incollection{Baumgartner1975:Ineffability,<br />
AUTHOR = {Baumgartner, James},<br />
TITLE = {Ineffability properties of cardinals. I},<br />
BOOKTITLE = {Infinite and finite sets &#40;Colloq., Keszthely, 1973; dedicated to P. Erd&#337;s on his 60th birthday), Vol. I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0384553 &#40;52 \#5427)},<br />
MRREVIEWER = {John K. Truss}<br />
}<br />
<br />
#Blass2010:CardinalCharacteristicsHandbook bibtex=@article{Blass2010:CardinalCharacteristicsHandbook,<br />
author = {Blass, Andreas},<br />
title = {Chapter 6: Cardinal characteristics of the continuum},<br />
journal = {Handbook of Set Theory},<br />
editor = {Foreman, Mathew; Kanamori, Akihiro},<br />
year = {2010},<br />
isbn = {1402048432},<br />
publisher = {Springer},<br />
url = {http://www.math.lsa.umich.edu/~ablass/hbk.pdf},<br />
}<br />
<br />
#Blass1976:ExactFunctors bibtex=@article{Blass1976:ExactFunctors,<br />
author = "Blass, Andreas",<br />
fjournal = "Pacific Journal of Mathematics",<br />
journal = "Pacific J. Math.",<br />
number = "2",<br />
pages = "335--346",<br />
publisher = "Pacific Journal of Mathematics, A Non-profit Corporation",<br />
title = "Exact functors and measurable cardinals.",<br />
url = "https://projecteuclid.org:443/euclid.pjm/1102867389",<br />
volume = "63",<br />
year = "1976"<br />
}<br />
<br />
#Boney2017:ModelTheoreticCharacterizations bibtex=@article{BBoney2017:ModelTheoreticCharacterizations,<br />
author = {Boney, Will},<br />
title = {Model Theoretic Characterizations of Large Cardinals},\<br />
year = {2017},<br />
eprint = {1708.07561},<br />
}<br />
<br />
#Bosch2006:SmallDefinablyLargeCardinals bibtex=@article {Bosch2006:SmallDefinablyLargeCardinals,<br />
AUTHOR = {Bosch, Roger},<br />
TITLE = {Small Definably-large Cardinals},<br />
JOURNAL = {Set Theory. Trends in Mathematics},<br />
YEAR = {2006},<br />
PAGES = {55-82},<br />
DOI = {10.1007/3-7643-7692-9_3},<br />
ISBN = {978-3-7643-7692-5},<br />
}<br />
<br />
#Cantor1915:ContributionsFoundingTransfinite bibtex=@book{Cantor1915:ContributionsFoundingTransfinite, <br />
author = {Cantor, Georg}, <br />
title = {Contributions to the Founding of the Theory of Transfinite Numbers}, <br />
editor = {Jourdain, Philip},<br />
note = {Original year was 1915}, <br />
publisher = {Dover}, <br />
address = {New York}, <br />
year = {1955}, <br />
isbn = {978-0-486-60045-1},<br />
url = {http://www.archive.org/details/contributionstot003626mbp},<br />
}<br />
<br />
#Carmody2015:ForceToChangeLargeCardinalStrength bibtex=@article{Carmody2015:ForceToChangeLargeCardinalStrength, <br />
author = {Carmody, Erin Kathryn}, <br />
title = {Force to change large cardinal strength}, <br />
year = {2015}, <br />
eprint = {1506.03432},<br />
url = {https://academicworks.cuny.edu/gc_etds/879/}<br />
}<br />
<br />
#CarmodyGitmanHabic2016:Mitchelllike bibtex=@article{CarmodyGitmanHabic2016:Mitchelllike, <br />
author = {Carmody, Erin and Gitman, Victoria and Habič, Miha E.}, <br />
title = {A Mitchell-like order for Ramsey and Ramsey-like cardinals}, <br />
year = {2016}, <br />
eprint = {1609.07645},<br />
}<br />
<br />
#CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal bibtex=@article{CodyGitikHamkinsSchanker2003:TheLeastWeaklyCompactCardinal, <br />
author = {Cody, Brent, Gitik, Moti, Hamkins, Joel David, and Schanker, Jason}, <br />
title = {The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact}, <br />
year = {2013}, <br />
eprint = {1305.5961},<br />
}<br />
<br />
#CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br />
title = "Easton's theorem for Ramsey and strongly Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "9",<br />
pages = "934 - 952",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "10.1016/j.apal.2015.04.006",<br />
url={https://victoriagitman.github.io/files/eastonramsey.pdf},<br />
AUTHOR= {Cody, Brent and Gitman, Victoria},<br />
}<br />
<br />
#Corazza2000:WholenessAxiomAndLaverSequences bibtex=@article{CorazzaAPAL,<br />
author = {Corazza, Paul},<br />
title = {The Wholeness Axiom and Laver sequences},<br />
journal = {Annals of Pure and Applied Logic},<br />
month={October},<br />
year = {2000},<br />
pages={157--260},<br />
}<br />
<br />
#Corazza2003:GapBetweenI3andWA bibtex=@ARTICLE{Corazza2003:GapBetweenI3andWA,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The gap between $\mathrm{I}_3$ and the wholeness axiom},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {179},<br />
YEAR = {2003},<br />
NUMBER = {1},<br />
PAGES = {43--60},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03E55 &#40;03E65)},<br />
MRNUMBER = {MR2028926 &#40;2004k:03100)},<br />
MRREVIEWER = {A. Kanamori},<br />
DOI = {10.4064/fm179-1-4},<br />
URL = {http://dx.doi.org/10.4064/fm179-1-4},<br />
}<br />
<br />
#Corazza2006:TheSpectrumOfElementaryEmbeddings bibtex=@ARTICLE{Corazza2006:TheSpectrumOfElementaryEmbeddings,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The spectrum of elementary embeddings $j : V \to V$},<br />
JOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {139},<br />
MONTH = {May},<br />
YEAR = {2006},<br />
NUMBER = {1--3},<br />
PAGES = {327-399},<br />
DOI = {10.1016/j.apal.2005.06.014},<br />
}<br />
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#Corazza2010:TheAxiomOfInfinityAndJVV bibtex=@ARTICLE{Corazza2010:TheAxiomOfInfinityAndJVV,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The Axiom of Infinity and transformations $j: V \to V$},<br />
JOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {1},<br />
PAGES = {37--84},<br />
DOI = {10.2178/bsl/1264433797},<br />
URL = {https://www.math.ucla.edu/~asl/bsl/1601/1601-002.ps},<br />
}<br />
<br />
#DaghighiPourmahdian2018:PropertiesShelah bibtex=@article{DaghighiPourmahdian2018:PropertiesShelah,<br />
AUTHOR = {Daghighi, Ali Sadegh and Pourmahdian, Massoud},<br />
TITLE = {On Some Properties of Shelah Cardinals},<br />
JOURNAL = {Bull. Iran. Math. Soc.},<br />
FJOURNAL = {Bulletin of the Iranian Mathematical Society},<br />
VOLUME = {44},<br />
YEAR = {2018},<br />
MONTH = {October},<br />
NUMBER = {5},<br />
PAGES = {1117-1124},<br />
DOI = {10.1007/s41980-018-0075-0},<br />
URL = {http://www.alidaghighi.org/wp-content/uploads/2017/08/On-Some-Properties-of-Shelah-Cardinals.pdf}<br />
}<br />
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#Dimonte2017:I0AndRankIntoRankAxioms bibtex=@article {Dimonte2017:I0AndRankIntoRankAxioms,<br />
AUTHOR = {Dimonte, Vincenzo},<br />
TITLE = {I0 and rank-into-rank axioms},<br />
YEAR = {2017},<br />
EPRINT = {1707.02613}<br />
}<br />
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#Dimopoulos2019:WoodinForStrongCompactness bibtex=@article {Dimopoulos2019:WoodinForStrongCompactness,<br />
title={Woodin for strong compactness cardinals},<br />
volume={84},<br />
DOI={10.1017/jsl.2018.67},<br />
number={1},<br />
journal={The Journal of Symbolic Logic},<br />
publisher={Cambridge University Press},<br />
author={Dimopoulos, Stamatis},<br />
year={2019},<br />
pages={301–319},<br />
eprint={1710.05743}<br />
}<br />
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#DoddJensen1982:CoreModel bibtex=@article {MR611394,<br />
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ISSN = {0003-4843},<br />
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}<br />
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#DonderKoepke1998:AccessibleJonsson bibtex=@article{DonderKoepke1983:AccessibleJonsson, <br />
author = {Donder, Hans-Dieter and Koepke, Peter},<br />
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year = {1998},<br />
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}<br />
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#DonderLevinski1989:PrinciplesRelatedChangsConjecture bibtex=@article{DonderLevinski1989:PrinciplesRelatedChangsConjecture, <br />
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title = {Some principles related to Chang's conjecture},<br />
journal = {Annals of Pure and Applied Logic},<br />
year = {1989},<br />
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doi = {10.1016/0168-0072&#40;89)90030-4},<br />
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}<br />
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AUTHOR = {Drake, Frank},<br />
PUBLISHER = {North-Holland Pub. Co.},<br />
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SERIES = {Studies in Logic and the Foundations of Mathematics, Volume 76}<br />
}<br />
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#Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals bibtex=@article{Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals,<br />
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year = {2005},<br />
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}<br />
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#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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ISSN = {0001-5954},<br />
MRCLASS = {04.60},<br />
MRNUMBER = {0141603 &#40;25 \#5001)},<br />
MRREVIEWER = {L. Gillman},<br />
}<br />
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#ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {On the structure of set-mappings},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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}<br />
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#EskrewHayut2016:LocalGlobalChangsConjecture bibtex=@article{EskrewHayut2016:LocalGlobalChangsConjecture,<br />
author = {Eskrew, Monroe and Hayut, Yair},<br />
title = {On the consistency of local and global versions of Chang's Conjecture},<br />
year = {2016},<br />
eprint = {1607.04904v4},<br />
}<br />
<br />
#Esser96:GPKAFA bibtex=@article{Esser96:GPKAFA,<br />
author = {Esser, Olivier},<br />
title = {Inconsistency of GPK+AFA},<br />
year = {1996},<br />
journal = {Mathematical Logic Quarterly},<br />
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pages = {104--108},<br />
url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19960420109/abstract}<br />
}<br />
<br />
#Esser96:InterpretationZFCandMKinPositiveTheory bibtex=@article{Esser96:InterpretationZFCandMKinPositiveTheory,<br />
author = {Esser, Olivier},<br />
title = {An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory},<br />
year = {1997},<br />
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url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19970430309/abstract}<br />
}<br />
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#Esser99:ConsistencyPositiveTheory bibtex=@article{Esser96:ConsistencyPositiveTheory,<br />
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title = {On the Consistency of a Positive Theory},<br />
year = {1999},<br />
journal = {Mathematical Logic Quarterly},<br />
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}<br />
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#Esser2000:InconsistencyACwithGPK bibtex=@article{Esser2000:InconsistencyACwithGPK,<br />
author = {Esser, Olivier},<br />
title = {Inconsistency of the Axiom of Choice with the Positive Theory $GPK^+_\infty$},<br />
year = {2000},<br />
month = {Dec.}<br />
journal = {Journal of Symbolic Logic},<br />
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number = {4},<br />
pages = {1911--1916},<br />
doi = {10.2307/2695086},<br />
url = {http://www.jstor.org/stable/2695086}<br />
}<br />
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#Esser99:ExtensionalityInPositiveTheory bibtex=@article{Esser96:ExtensionalityInPositiveTheory,<br />
author = {Esser, Olivier},<br />
title = {On the axiom of extensionality in the positive set theory},<br />
year = {2003},<br />
journal = {Mathematical Logic Quarterly},<br />
doi = {10.1002/malq.200310009},<br />
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url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.200310009/abstract}<br />
}<br />
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#EvansHamkins:TransfiniteGameValuesInInfiniteChess bibtex=@ARTICLE{EvansHamkins:TransfiniteGameValuesInInfiniteChess,<br />
AUTHOR = {Evans, C. D. A. and Hamkins, Joel David},<br />
TITLE = {Transfinite game values in infinite chess},<br />
JOURNAL = {},<br />
YEAR = {},<br />
volume = {},<br />
number = {},<br />
pages = {},<br />
month = {},<br />
note = {under review},<br />
eprint = {1302.4377},<br />
url = {http://jdh.hamkins.org/game-values-in-infinite-chess},<br />
abstract = {},<br />
keywords = {},<br />
source = {},<br />
}<br />
<br />
#Feng1990:HierarchyRamsey bibtex=@article{Feng1990:HierarchyRamsey,<br />
title = "A hierarchy of Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "49",<br />
number = "3",<br />
pages = "257 - 277",<br />
year = "1990",<br />
issn = "0168-0072",<br />
doi = "10.1016/0168-0072&#40;90)90028-Z",<br />
author = "Feng, Qi",<br />
}<br />
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#Foreman2010:Handbook bibtex=@book<br />
{Foreman2010:Handbook, <br />
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editor = {Foreman, Matthew and Kanamori, Akihiro}, <br />
title = {Handbook of Set Theory},<br />
edition = {First}, <br />
publisher = {Springer}, <br />
year = {2010}, <br />
isbn = {978-1-4020-4843-2},<br />
note = {This book is actually a compendium of articles from multiple authors},<br />
url = {http://www.springer.com/mathematics/book/978-1-4020-4843-2},<br />
}<br />
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AUTHOR = {Forti, M and Hinnion, R.},<br />
TITLE = {The Consistency Problem for Positive Comprehension Principles},<br />
JOURNAL = {J. Symbolic Logic},<br />
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NUMBER = {4},<br />
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}<br />
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#Friedman1998:Subtle bibtex=@article{Friedman1998:Subtle,<br />
AUTHOR = {Friedman, Harvey M.},<br />
TITLE = {Subtle cardinals and linear orderings},<br />
YEAR = {1998},<br />
URL = {https://u.osu.edu/friedman.8/files/2014/01/subtlecardinals-1tod0i8.pdf}<br />
}<br />
<br />
#FuchsHamkinsReitz2015:SetTheoreticGeology bibtex=@article{FuchsHamkinsReitz2015:SetTheoreticGeology<br />
title = "Set-theoretic geology",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "4",<br />
pages = "464 - 501",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "https://doi.org/10.1016/j.apal.2014.11.004",<br />
url = "http://www.sciencedirect.com/science/article/pii/S0168007214001225",<br />
author = "Fuchs, Gunter and Hamkins, Joel David and Reitz, Jonas",<br />
title = "Set-theoretic geology",<br />
eprint = "1107.4776",<br />
}<br />
<br />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory &#40;Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 &#40;02H13)},<br />
MRNUMBER = {0376347 &#40;51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo bibtex=@article{GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo,<br />
AUTHOR = {Gitman, Victoria and Hamkins, Joel David},<br />
TITLE = {A model of the generic Vopěnka principle in which the ordinals are not Mahlo},<br />
YEAR = {2018},<br />
EPRINT = {1706.00843v2}<br />
}<br />
<br />
#GitmanJohnstone:IndestructiblyRamsey bibtex=@ARTICLE{GitmanJohnstone:IndestructiblyRamsey,<br />
AUTHOR= {Gitman, Victoria and Johnstone, Thomas A.},<br />
TITLE= {Indestructibility for Ramsey and Ramsey-like cardinals},<br />
NOTE= {In preparation},<br />
URL= {https://victoriagitman.github.io/files/indestructibleramseycardinalsnew.pdf}<br />
}<br />
<br />
#GitmanSchindler:VirtualLargeCardinals bibtex=@ARTICLE{GitmanSchindler:VirtualLargeCardinals,<br />
AUTHOR= {Gitman, Victoria and Shindler, Ralf},<br />
TITLE= {Virtual large cardinals},<br />
URL= {https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf}<br />
}<br />
<br />
#Goldblatt1998: bibtex=@book{Goldblatt1998:ultrafilter,<br />
AUTHOR = {Goldblatt, Robert},<br />
TITLE = {Lectures on the Hyperreals},<br />
PUBLISHER = {Springer},<br />
YEAR = {1998},<br />
}<br />
<br />
#GoldsternShelah1995:BPFA bibtex = @article{GoldsternShelah1995:BPFA,<br />
AUTHOR = {Goldstern, Martin and Shelah, Saharon},<br />
TITLE = {The Bounded Proper Forcing Axiom},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#Golshani2017:EastonLikeInPresenceShelah bibtex=@article{Golshani2017:EastonLikeInPresenceShelah,<br />
AUTHOR = {Golshani, Mohammad},<br />
TITLE = {An Easton like theorem in the presence of Shelah cardinals},<br />
JOURNAL = {M. Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
MONTH = {May},<br />
NUMBER = {3-4},<br />
PAGES = {273-287},<br />
DOI = {10.1007/s00153-017-0528-9},<br />
URL = {http://math.ipm.ac.ir/~golshani/Papers/An%20Easton%20like%20theorem%20in%20the%20presence%20of%20Shelah%20Cardinals.pdf}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article{HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {1771072 &#40;2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 &#40;03E65)},<br />
MRNUMBER = {1816602 &#40;2001m:03102)},<br />
MRREVIEWER = {Ralf-Dieter Schindler},<br />
DOI = {10.1007/s001530050169},<br />
URL = {http://dx.doi.org/10.1007/s001530050169},<br />
eprint = {math/9902079},<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
<!--file = F,--><!--Is it important? It causes an error.--><br />
}<br />
<br />
#Hamkins2008:UnfoldableGCH bibtex=@article{Hamkins2008:UnfoldableGCH, <br />
author = {Hamkins, Joel David},<br />
title = {Unfoldable cardinals and the GCH},<br />
year = {2008},<br />
eprint={math/9909029},<br />
}<br />
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#Hamkins2009:TallCardinals bibtex=@ARTICLE{Hamkins2009:TallCardinals,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Tall cardinals},<br />
JOURNAL = {MLQ Math. Log. Q.},<br />
FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br />
VOLUME = {55},<br />
YEAR = {2009},<br />
NUMBER = {1},<br />
PAGES = {68--86},<br />
ISSN = {0942-5616},<br />
MRCLASS = {03E55 &#40;03E35)},<br />
MRNUMBER = {2489293 &#40;2010g:03083)},<br />
MRREVIEWER = {Carlos A. Di Prisco},<br />
DOI = {10.1002/malq.200710084},<br />
URL = {http://boolesrings.org/hamkins/tallcardinals/},<br />
}<br />
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#HamkinsJohnstone2010:IndestructibleStrongUnfoldability bibtex=@article{HamkinsJohnstone2010:IndestructibleStrongUnfoldability,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Indestructible strong un-foldability},<br />
YEAR = {2010},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
VOLUME = {51},<br />
NUMBER = {3},<br />
PAGES = {291--321}<br />
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#HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory bibtex=@article{HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory,<br />
author = {Hamkins, Joel David; Linetsky, David; Reitz, Jonas},<br />
title = {Pointwise Definable Models of Set Theory},<br />
year = {2012},<br />
eprint = {1105.4597}<br />
}<br />
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#HamkinsJohnstone:ResurrectionAxioms bibtex=@article{HamkinsJohnstone:ResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Resurrection axioms and uplifting cardinals},<br />
YEAR = {2014},<br />
url = {http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals/},<br />
eprint = {1307.3602},<br />
}<br />
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#HamkinsJohnstone:BoldfaceResurrectionAxioms bibtex=@article{HamkinsJohnstone:BoldfaceResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Strongly uplifting cardinals and the boldface resurrection axioms},<br />
YEAR = {2014},<br />
eprint = {1403.2788},<br />
}<br />
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#Hamkins2016:TheVopenkaPrincipleIs bibtex=@article{Hamkins2016:TheVopenkaPrincipleIs,<br />
author = {Hamkins, Joel David},<br />
title = {The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme},<br />
year = {2016},<br />
url = {http://jdh.hamkins.org/vopenka-principle-vopenka-scheme/}<br />
eprint = {1606.03778},<br />
}<br />
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#Hauser1991:IndescribableElementaryEmbeddings bibtex=@article{<br />
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#HolySchlicht2017:HierarchyRamseylike bibtex=@article{HolySchlicht2017:HierarchyRamseylike, <br />
author = {Holy, Peter and Schlicht, Philipp}, <br />
title = {A hierarchy of Ramsey-like cardinals}, <br />
year = {2018},<br />
eprint = {1710.10043},<br />
doi = {10.4064/fm396-9-2017},<br />
journal = {Fundamenta Mathematicae},<br />
volume = {242},<br />
pages = {49-74},<br />
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}<br />
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#JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson bibtex=@article{<br />
JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson,<br />
AUTHOR = {Jackson, Steve; Ketchersid, Richard; Schlutzenberg, Farmer; Woodin, W. Hugh},<br />
TITLE = {Determinacy and Jónsson cardinals in $L&#40;\mathbb{R})$},<br />
YEAR = {2015},<br />
DOI = {10.1017/jsl.2014.49},<br />
EPRINT = {1304.2323}<br />
}<br />
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#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
SERIES = {Springer Monographs in Mathematics},<br />
NOTE = {The third millennium edition, revised and expanded},<br />
PUBLISHER = {Springer-Verlag},<br />
EDITION = {Third},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<br />
URL = {https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf},<br />
}<br />
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#JensenKunen1969:Ineffable bibtex=@unpublished{JensenKunen1969:Ineffable,<br />
AUTHOR={Jensen, Ronald and Kunen, Kenneth},<br />
TITLE={Some combinatorial properties of $L$ and $V$},<br />
YEAR={1969},<br />
URL={http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html},<br />
}<br />
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#Kanamori1977:EvolutionLargeCardinals bibtex=@incollection {#Kanamori1977:EvolutionLargeCardinals,<br />
AUTHOR = {Kanamori, Akihiro and Magidor, Menachem},<br />
TITLE = {The evolution of large cardinal axioms in set theory},<br />
BOOKTITLE = {Higher set theory &#40;Proc. Conf., Math. Forschungsinst.,<br />
Oberwolfach, 1977)},<br />
SERIES = {Lecture Notes in Math.},<br />
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PAGES = {99--275},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1978},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {520190 &#40;80b:03083)},<br />
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{Kanamori1978:StrongAxioms, <br />
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note = {In ''Annals of Mathematical Logic'', '''13'''&#40;1978)}, <br />
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#KanamoriAwerbuchFriedlander1990:Compleat0Dagger bibtex=@article{KanamoriAwerbuchFriedlander1990:Compleat0Dagger,<br />
author = {Kanamori, Akihiro and Awerbuch-Friedlander, Tamara},<br />
title = {The compleat 0†},<br />
journal = {Mathematical Logic Quarterly},<br />
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year = {1990}<br />
}<br />
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#Kanamori2009:HigherInfinite bibtex=@book{Kanamori2009:HigherInfinite,<br />
AUTHOR = {Kanamori, Akihiro},<br />
TITLE = {The higher infinite},<br />
SERIES = {Springer Monographs in Mathematics},<br />
EDITION = {Second},<br />
NOTE = {Large cardinals in set theory from their beginnings,<br />
Paperback reprint of the 2003 edition},<br />
PUBLISHER = {Springer-Verlag},<br />
ADDRESS = {Berlin},<br />
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PAGES = {xxii+536},<br />
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#Kentaro2007:DoubleHelix bibtex=@article{Kentaro2007:DoubleHelix,<br />
AUTHOR = {Kentaro, Sato},<br />
TITLE = {Double helix in large large cardinals and iteration of elementary embeddings},<br />
JOURNAL = {Annals of Pure and Applied Logic},<br />
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#Ketonen1974:SomeCombinatorialPrinciples bibtex=@article{Ketonen1974:SomeCombinatorialPrinciples,<br />
AUTHOR = {Ketonen, Jussi},<br />
TITLE = {Some combinatorial principles},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {188},<br />
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#KoellnerWoodin2010:LCFD bibtex=@article{KoellnerWoodin2010:LCFD,<br />
author = {Koellner, Peter and Woodin, W. Hugh},<br />
title = {Chapter 23: Large cardinals from Determinacy},<br />
journal = {Handbook of Set Theory},<br />
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year = {2010},<br />
publisher = {Springer},<br />
url = {http://logic.harvard.edu/koellner/LCFD.pdf}<br />
}<br />
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#Kunen1978:SaturatedIdeals bibtex=@article{Kunen1978:SaturatedIdeals,<br />
AUTHOR = {Kunen, Kenneth},<br />
TITLE = {Saturated Ideals},<br />
YEAR = {1978},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
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#Larson2010:HistoryDeterminacy bibtex=@article{<br />
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TITLE = {A brief history of determinacy},<br />
YEAR = {2013},<br />
URL = {http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf}<br />
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#Laver1997:Implications bibtex=@article {Laver1997:Implications,<br />
AUTHOR = {Laver, Richard},<br />
TITLE = {Implications between strong large cardinal axioms},<br />
JOURNAL = {Ann. Math. Logic},<br />
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MRCLASS = {03E55 &#40;03E35)},<br />
MRNUMBER = {1489305 &#40;99c:03074)},<br />
MRREVIEWER = {Douglas R. Burke},<br />
}<br />
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#Leshem2000:OCDefinableTreeProperty bibtex=@article {Leshem2000:OCDefinableTreeProperty,<br />
AUTHOR = {Leshem, Amir},<br />
TITLE = {On the consistency of the definable tree property on $\aleph_1$},<br />
JOURNAL = {J. Symbolic Logic},<br />
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DOI = {10.2307/2586696},<br />
EPRINT = {math/0005208},<br />
}<br />
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#Maddy88:BelAxiomsI bibtex=@article{Maddy88:BelAxiomsI,<br />
AUTHOR = {Maddy, Penelope},<br />
TITLE = {Believing the axioms. I},<br />
JOURNAL = {J. Symbolic Logic},<br />
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#Maddy88:BelAxiomsII bibtex=@article{Maddy88:BelAxiomsII,<br />
AUTHOR = {Maddy, Penelope},<br />
TITLE = {Believing the axioms. II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
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#Madore2017:OrdinalZoo bibtex=@article{Madore2017:OrdinalZoo,<br />
AUTHOR = {Madore, David},<br />
TITLE = {A zoo of ordinals},<br />
YEAR = {2017},<br />
URL = {http://www.madore.org/~david/math/ordinal-zoo.pdf}<br />
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#Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br />
AUTHOR = {Makowsky, Johann},<br />
TITLE = {Vopěnka's Principle and Compact Logics},<br />
JOURNAL = {J. Symbol Logic},<br />
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YEAR = {1985},<br />
}<br />
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#Marek1974:StableSets bibtex=@article{Marek1974:StableSets,<br />
author = {Marek, W.},<br />
journal = {Fundamenta Mathematicae},<br />
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pages = {175-189},<br />
title = {Stable sets, a characterization of $β_2$-models of full second order arithmetic and some related facts},<br />
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}<br />
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AUTHOR = {Mitchell, William J.},<br />
TITLE = {Jónsson Cardinals, Erdős Cardinals, and the Core Model},<br />
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#Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br />
AUTHOR = {Mitchell, William J.},<br />
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#Miyamoto1998:ANoteOnWeakSegmentsOfPFA bibtex=@article{Miyamoto1998:ANoteOnWeakSegmentsOfPFA,<br />
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#NielsenWelch2018:GamesRamseylike bibtex=@article{NielsenWelch2018:GamesRamseylike, <br />
author = {Nielsen, Dan Saattrup and Welch, Philip}, <br />
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}<br />
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#Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge bibtex=@article{Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge,<br />
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#RichterAczel1974:InductiveDefinitions bibtex=@incollection{RichterAczel1974:InductiveDefinitions,<br />
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#Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br />
AUTHOR = {Schanker, Jason A.},<br />
TITLE = {Partial near supercompactness},<br />
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#Schanker2011:WeaklyMeasurableCardinals bibtex=@article{Schanker2011:WeaklyMeasurableCardinals,<br />
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#Schimmerling2002:WoodinShelahAndCoreModel bibtex=@article {Schimmerling2002:WoodinShelahAndCoreModel,<br />
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#Schindler2000:RemarkableCardinal bibtex=@article {Schindler2000:RemarkableCardinal,<br />
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PAGES = {45--110},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35},<br />
MRNUMBER = {0409188 &#40;53 \#12950)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
#Suzuki1998:NojVtoVinVofG bibtex=@article{Suzuki1998:NojVtoVinVofG,<br />
AUTHOR = {Suzuki, Akira},<br />
TITLE = {Non-existence of generic elementary embeddings into the ground<br />
model},<br />
JOURNAL = {Tsukuba J. Math.},<br />
FJOURNAL = {Tsukuba Journal of Mathematics},<br />
VOLUME = {22},<br />
YEAR = {1998},<br />
NUMBER = {2},<br />
PAGES = {343--347},<br />
ISSN = {0387-4982},<br />
MRCLASS = {03E55 &#40;03E05)},<br />
MRNUMBER = {MR1650737 &#40;2000a:03087)},<br />
Abstract = {The author proves that if $j\colon V\rightarrow M$ is an elementary embedding defined in a set generic extension of $V$, then $V \not \subseteq M$. The proof generalizes Woodin's proof of Kunen's theorem to generic embeddings.},<br />
MRREVIEWER = {Douglas R. Burke},<br />
}<br />
<br />
#Suzuki1999:NoDefinablejVtoVinZF bibtex=@article{Suzuki1999:NoDefinablejVtoVinZF,<br />
AUTHOR = {Suzuki, Akira},<br />
TITLE = {No elementary embedding from $V$ into $V$ is definable<br />
from parameters},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {64},<br />
YEAR = {1999},<br />
NUMBER = {4},<br />
PAGES = {1591--1594},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E47},<br />
MRNUMBER = {MR1780073 &#40;2002h:03114)},<br />
DOI = {10.2307/2586799},<br />
URL = {http://dx.doi.org/10.2307/2586799},<br />
}<br />
<br />
#TrangWilson2016:DetFromStrongCompactness bibtex=@article{TrangWilson2016:DetFromStrongCompactness,<br />
AUTHOR = {Trang, Nam and Wilson, Trevor},<br />
TITLE = {Determinacy from Strong Compactness of $\omega_1$},<br />
YEAR = {2016},<br />
EPRINT = {1609.05411v1}<br />
}<br />
<br />
#TrybaJan1983:JonssonUncountable bibtex=@article{TrybaJan1983:JonssonUncountable,<br />
AUTHOR = {Tryba, Jan},<br />
TITLE = {On Jónsson cardinals with uncountable cofinality},<br />
YEAR = {1983},<br />
JOURNAL = {Israel Journal of Mathematics},<br />
VOLUME = {49},<br />
NUMBER = {4},<br />
}<br />
<br />
#Usuba2017:DDGandVeryLarge bibtex=@article{Usuba2017:DDGandVeryLarge,<br />
author = {Usuba, Toshimichi},<br />
title = {The downward directed grounds hypothesis and very large cardinals},<br />
year = {2017},<br />
eprint = {1707.05132},<br />
doi = {10.1142/S021906131750009X},<br />
journal = {Journal of Mathematical Logic},<br />
volume = {17},<br />
number = {02},<br />
pages = {1750009},<br />
issn = {0219-0613},<br />
publisher = {World Scientific Publishing Co. Pte Ltd},<br />
}<br />
<br />
#Usuba2018:ExtendibleCardinalsAndTheMantle bibtex=@article{Usuba2018:ExtendibleCardinalsAndTheMantle,<br />
author = {Usuba, Toshimichi},<br />
title = {Extendible cardinals and the mantle},<br />
year = {2019},<br />
eprint = {1803.03944},<br />
doi = {10.1007/s00153-018-0625-4},<br />
journal = {Archive for Mathematical Logic},<br />
volume = {58},<br />
number = {1-2},<br />
pages = {71-75},<br />
}<br />
<br />
#VialeWeiss2011:OnConsistencyStrengthPFA bibtex=@article{VialeWeiss2011:OnConsistencyStrengthPFA,<br />
AUTHOR = {Viale, Matteo and Weiß, Christoph},<br />
TITLE = {On the consistency strength of the proper forcing axiom},<br />
JOURNAL = {Advances in Mathematics},<br />
VOLUME = {228},<br />
YEAR = {2011},<br />
NUMBER = {5},<br />
PAGES = {2672--2687},<br />
EPRINT = {1012.2046},<br />
MRCLASS = {03E57 &#40;03E05 03E55)},<br />
MRNUMBER = {MR2838054 &#40;2012m:03131)},<br />
}<br />
<br />
#Villaveces1996:ChainsEndElementaryExtensionsModels bibtex=@article{Villaveces1996:ChainsEndElementaryExtensionsModels,<br />
AUTHOR = {Villaveces, Andrés},<br />
TITLE = {Chains of End Elementary Extensions of Models of Set Theory},<br />
JOURNAL = {JSTOR},<br />
YEAR = {1996},<br />
EPRINT = {math/9611209},<br />
}<br />
<br />
#Welch1998:InnerModels bibtex=@article{Welch1998:InnerModels,<br />
author = {Welch, Philip},<br />
title = {Some remarks on the maximality of Inner Models},<br />
journal = {Logic Colloquium},<br />
year = {1998},<br />
url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.41.7037&rep=rep1&type=pdf},<br />
}<br />
<br />
#Welch2000:LengthsOfITTM bibtex=@article{Welch2000:LengthsOfITTM,<br />
author={Welch, Philip},<br />
title = {The Lengths of Infinite Time Turing Machine Computations},<br />
journal = {Bulletin of the London Mathematical Society},<br />
volume = {32},<br />
number = {2},<br />
pages = {129--136},<br />
year = {2000},<br />
}<br />
<br />
#Wilson2018:WeaklyRemarkableCardinals bibtex=@article{Wilson2018:WeaklyRemarkableCardinals,<br />
AUTHOR = {Wilson, Trevor M.},<br />
TITLE = {Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle},<br />
YEAR = {2018},<br />
EPRINT = {1807.02207v1}<br />
}<br />
<br />
#Woodin2010:SEM1 bibtex=@article{doi:10.1142/S021906131000095X,<br />
author = {Woodin, W. Hugh},<br />
title = {Suitable extender models I},<br />
journal = {Journal of Mathematical Logic},<br />
volume = {10},<br />
number = {01n02},<br />
pages = {101-339},<br />
year = {2010},<br />
doi = {10.1142/S021906131000095X},<br />
URL = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br />
}<br />
<br />
#Woodin2011:SEM2 bibtex=@article{doi:10.1142/S021906131100102X,<br />
author = {Woodin, W. Hugh},<br />
title = {Suitable extender models II: beyond $\omega$-huge},<br />
journal = {Journal of Mathematical Logic},<br />
volume = {11},<br />
number = {02},<br />
pages = {115-436},<br />
year = {2011},<br />
doi = {10.1142/S021906131100102X},<br />
URL = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131100102X}<br />
}<br />
<br />
#Zapletal1996:ANewProofOfKunenInconsistency bibtex=@article {Zapletal1996:ANewProofOfKunenInconsistency,<br />
AUTHOR = {Zapletal, Jindrich},<br />
TITLE = {A new proof of Kunen's inconsistency},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {124},<br />
YEAR = {1996},<br />
NUMBER = {7},<br />
PAGES = {2203--2204},<br />
ISSN = {0002-9939},<br />
CODEN = {PAMYAR},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {MR1317054 &#40;96i:03051)},<br />
MRREVIEWER = {L. Bukovsky},<br />
DOI = {10.1090/S0002-9939-96-03281-9},<br />
URL = {http://dx.doi.org/10.1090/S0002-9939-96-03281-9},<br />
}<br />
</biblio><br />
<br />
== User instructions ==<br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>BartekChomhttp://cantorsattic.info/index.php?title=Unfoldable&diff=4164Unfoldable2022-05-29T12:28:17Z<p>BartekChom: == Relation to forcing == From <cite>Hamkins2008:UnfoldableGCH</cite>:</p>
<hr />
<div>{{DISPLAYTITLE: Unfoldable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
The unfoldable cardinals were introduced by Andres Villaveces in order to generalize the definition of [[weakly compact|weak compactness]]. Because weak compactness has many different definitions, the one he chose to extend was specifically the embedding property &#40;see weakly compact for more information). The way he did this was analogous to the generalization of [[huge]] cardinals to superhuge cardinals.<br />
<br />
== Definition ==<br />
<br />
There are unfoldable cardinals and strongly unfoldable cardinals, as well as superstrongly unfoldable &#40;AKA almost-hugely unfoldable AKA [[uplifting|strongly uplifting]]) cardinals. All of these are generalizations of [[weakly compact|weak compactness]].<br />
<br />
=== Unfoldable ===<br />
<br />
A cardinal $\kappa$ is '''$\theta$-unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j&#40;\kappa)\geq\theta$. $\kappa$ is then called '''unfoldable''' iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.<br />
<br />
Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=&#40;M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $&#40;\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=&#40;N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:<br />
<br />
*$\mathcal{M}$ is an elementary substructure of $\mathcal{N}$<br />
*$\mathcal{M}\neq\mathcal{N}$<br />
*For any $a\in M$, $b\in N$, and $\gamma<\beta$, $b R_\gamma^\mathcal{N} a\rightarrow b\in M$<br />
<br />
If such holds, $\mathcal{M}$ is said to be '''nontrivially end elementary extended''' by $\mathcal{N}$ and '''$\mathcal{N}$ is a nontrivial end elementary extension''' of $\mathcal{M}$, abbreviated $\mathcal{N}$ is an '''eee''' of $\mathcal{M}$.<br />
<br />
A cardinal $\kappa$ is '''$\lambda$-unfoldable''' iff $\kappa$ is [[inaccessible]] and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $&#40;V_\kappa;\in,S)$ such that $M\not\in V_\lambda$. $\kappa $ is '''unfoldable''' iff $M $ can be made to have arbitrarily large rank. In this case, $&#40;V_\kappa;\in,S)\prec_e &#40;M;\in^M,S^M)$ iff $&#40;V_\kappa;\in,S)\prec &#40;M;\in^M,S^M)$ and $&#40;V_\kappa;\in)\prec_e &#40;M;\in^M)$. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
$\kappa$ is also '''unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $&#40;V_\kappa;\in,S)$, $&#40;\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $&#40;V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
=== Long Unfoldable ===<br />
<br />
$\kappa$ is '''long unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $&#40;V_\kappa;\in,S)$, $&#40;\mathcal{E};\prec_e)$ has chains of length $\text{Ord}$. <br />
<br />
Every long unfoldable cardinal is unfoldable. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
<br />
=== Strongly Unfoldable ===<br />
<br />
A cardinal $\kappa$ is '''$\theta$-strongly unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j&#40;\kappa)\geq\theta$ and $V_\theta\subseteq N$.<br />
<br />
$\kappa$ is then called '''strongly unfoldable''' iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.<br />
<br />
As defined in <cite>HamkinsJohnstone2010:IndestructibleStrongUnfoldability</cite> in analogy with [[Mitchell rank]]s, a strongly unfoldable cardinal $\kappa$ is '''strongly unfoldable of degree $\alpha$''', for an ordinal $\alpha$, if for every ordinal $\theta$ it is $\theta$-strongly unfoldable of degree $\alpha$, meaning that for each $A \in H_{\kappa^+}$ there is a $\kappa$-[[model]] $M \models \mathrm{ZFC}$ with $A \in M$ and a transitive set $N$ with $\alpha \in M$ and an elementary embedding $j:M \to N$ having critical point $\kappa$ with $j&#40;\kappa)>\max\{\theta, \alpha\}$ and $V_\theta \subset N$, such that $\kappa$ is strongly unfoldable of every degree $\beta < \alpha$ in $N$.<cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite><br />
<br />
The strongly unfoldable cardinals are exactly the [[shrewd]] cardinals. [https://arxiv.org/abs/2107.12722]<br />
<br />
=== Superstrongly Unfoldable ===<br />
<br />
Superstrongly unfoldable and almost-hugely unfoldable cardinals are defined and shown to be equivalent to [[uplifting|strongly uplifting]] &#40;described there) in <cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite>.<br />
<br />
== Relations to Other Cardinals ==<br />
Here is a list of relations between unfoldability and other large cardinal axioms:<br />
<br />
*For every finite $m$ and $n$, unfoldability implies the consistency of the existence of a [[indescribable|$\Pi_m^n$-indescribable]] cardinal &#40;specifically, such cardinals exist in $V_\kappa\cap L$ for some $\kappa$). Furthermore, if $V=L$, then the least $\Pi_m^n$-indescribable cardinal is less than the least unfoldable cardinal, and every unfoldable cardinal is totally indescribable. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Any strongly unfoldable cardinal is [[indescribable|totally indescribable]] and a limit of totally indescribable cardinals. Therefore the consistency strength of unfoldability is stronger than total indescribability. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Every superstrongly unfoldable cardinal &#40;i.e. [[uplifting|strongly uplifting]] cardinal) is strongly unfoldable of every ordinal degree <math>\alpha</math>, and a stationary limit of cardinals that are strongly unfoldable of every ordinal degree and so on. <cite>HamkinsJohnstone:BoldfaceResurrectionAxioms</cite><br />
*The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable, which are precisely those that are [[weakly compact]]. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. <cite>Hamkins2008:UnfoldableGCH</cite><br />
*The assertion that a [[Ramsey]] cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Although unfoldable cardinals are consistency-wise stronger than [[weakly compact]] cardinals, if there is a strongly unfoldable cardinal, then in a forcing extension, the least weakly compact cardinal is also the least unfoldable cardinal.<cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*The existence of a [[subtle]] cardinal is consistency-wise stronger than the existence of an unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*If a [[subtle]] cardinal and an unfoldable cardinal exist and $V=L$, then the least unfoldable cardinal is larger than the least subtle cardinal &#40;and therefore much larger than the least [[weakly compact]]). <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Any [[Ramsey]] cardinal is unfoldable. If there is a weakly compact cardinal above an [[Erdos|$\omega_1$-Erdos]] cardinal, then the least unfoldable is less than that &#40;therefore less than the least Ramsey). <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite><br />
*Even though it may seem odd at first, if both exist then the least [[Rank-into-rank|I3]] cardinal is less than the least strongly unfoldable cardinal.<br />
* $ω_1$-[[iterable]] cardinals are strongly unfoldable in $L$.<cite>GitmanWelch2011:RamseyLikeCardinalsII</cite><br />
<br />
== Relation to forcing ==<br />
From <cite>Hamkins2008:UnfoldableGCH</cite>:<br />
* If $κ$ is unfoldable, then in a forcing extension by forcing of size $κ$ the unfoldability of $κ$ is indestructible by $\mathrm{Add}&#40;κ, θ)$ for any $θ$. &#40;Main Theorem 2)<br />
* If $κ$ is strongly unfoldable, then in a forcing extension by forcing of size $κ$ the strong unfoldability of $κ$ is indestructible by $\mathrm{Add}&#40;κ, 1)$ but not by $\mathrm{Add}&#40;κ, θ)$ for any cardinal $θ > κ$. &#40;Theorem 7)<br />
* Any unfoldable cardinal can be made, by forcing of size $κ^+$, $κ$ unfoldable but not strongly unfoldable. &#40;Corollary 8)<br />
* In a forcing extension preserving all strongly unfoldable cardinals, GCH fails at every [[inaccessible]] cardinal. &#40;Theorem 9)<br />
* Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every inaccessible cardinal. Therefore, GCH can fail at every unfoldable cardinal.<br />
<br />
''TODO: connection to weak forms of PFA''<br />
<br />
''TODO: consistency with slim Kurepa trees''<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Weakly_compact&diff=4163Weakly compact2022-05-15T13:38:01Z<p>BartekChom: \implies!</p>
<hr />
<div>{{DISPLAYTITLE: Weakly compact cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
Weakly compact cardinals lie at the focal point of a number<br />
of diverse concepts in infinite combinatorics, admitting various characterizations in terms of these concepts. If $\kappa^{{<}\kappa} = \kappa$, then the following are equivalent: <br />
<br />
:; Weak compactness : A cardinal $\kappa$ is weakly compact if and only if it is [[uncountable]] and every $\kappa$-satisfiable theory in an [[Infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] language of size at most $\kappa$ is satisfiable.<br />
:; Extension property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset V_\kappa$, there is a transitive structure $W$ properly extending $V_\kappa$ and $A^*\subset W$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.<br />
:; Tree property : A cardinal $\kappa$ is weakly compact if and only if it is [[inaccessible]] and has the [[tree property]].<br />
:; Filter property : A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a set containing at most $\kappa$-many subsets of $\kappa$, then there is a $\kappa$-[[filter|complete nonprincipal filter]] $F$ measuring every set in $M$.<br />
:; Weak embedding property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with $\kappa\in M$ and a transitive set $N$ with an [[elementary embedding|embedding]] $j:M\to N$ with [[critical point]] $\kappa$.<br />
:; Embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a transitive set $N$ and an embedding $j:M\to N$ with critical point $\kappa$.<br />
:; Normal embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$, such that $N=\{\ j&#40;f)&#40;\kappa)\mid f\in M\ \}$.<br />
;; Hauser embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.<br />
:; Partition property : A cardinal $\kappa$ is weakly compact if and only if the [[partition property]] $\kappa\to&#40;\kappa)^2_2$ holds.<br />
:; Indescribability property : A cardinal $\kappa$ is weakly compact if and only if it is $\Pi_1^1$-[[indescribable]].<br />
:; Skolem Property : A cardinal $\kappa$ is weakly compact if and only if $\kappa$ is inaccessible and every $\kappa$-unboundedly satisfiable $\mathcal{L}_{\kappa,\kappa}$-theory $T$ of size at most $\kappa$ has a model of size at least $\kappa$. A theory $T$ is $\kappa$-unboundedly satisfiable if and only if for any $\lambda<\kappa$, there exists a model $\mathcal{M}\models T$ with $\lambda\leq|M|<\kappa$. For more info see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937#309937 here].<br />
:; 2-regular : A cardinal $\kappa$ is weakly compact if and only if every $\kappa$-bounded $F: \kappa_\kappa\rightarrow\kappa_\kappa$ has a witness &#40;$0<\alpha<\kappa$ such that for every $f: \kappa\rightarrow\kappa$ we have $f|\alpha\subseteq\alpha \implies F(f)"\alpha\subseteq\alpha$). ''TODO complete'' <cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.13<!--typo, written as 1.3-->, theorem 1.14</sup><br />
<br />
Weakly compact cardinals first arose<br />
in connection with &#40;and were named for) the question of<br />
whether certain [[Infinitary logic|infinitary logics]] satisfy the compactness<br />
theorem of first order logic. Specifically, in a language<br />
with a signature consisting, as in the first order context,<br />
of a set of constant, finitary function and relation<br />
symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$<br />
formulas by closing the collection of formulas under<br />
infinitary conjunctions<br />
$\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions<br />
$\vee_{\alpha<\delta}\varphi_\alpha$ of any size<br />
$\delta<\kappa$, as well as infinitary quantification<br />
$\exists\vec x$ and $\forall\vec x$ over blocks of<br />
variables $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less<br />
than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics.<br />
A theory is ''$\theta$-satisfiable'' if every subtheory<br />
consisting of fewer than $\theta$ many sentences of it is<br />
satisfiable. First order logic is precisely<br />
$L_{\omega,\omega}$, and the classical Compactness theorem<br />
asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$<br />
theory is satisfiable. A uncountable cardinal $\kappa$ is<br />
''[[strongly compact]]'' if every $\kappa$-satisfiable<br />
$\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal<br />
$\kappa$ is ''weakly compact'' if every<br />
$\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a<br />
language having at most $\kappa$ many constant, function<br />
and relation symbols, is satisfiable.<br />
<br />
Next, for any cardinal $\kappa$, a ''$\kappa$-tree'' is a<br />
tree of height $\kappa$, all of whose levels have size less<br />
than $\kappa$. More specifically, $T$ is a ''tree'' if<br />
$T$ is a partial order such that the predecessors of any<br />
node in $T$ are well ordered. The $\alpha^\textrm{th}$ level of a<br />
tree $T$, denoted $T_\alpha$, consists of the nodes whose<br />
predecessors have order type exactly $\alpha$, and these<br />
nodes are also said to have ''height'' $\alpha$. The height of the tree $T$ is the first $\alpha$ for which $T$<br />
has no nodes of height $\alpha$. A ""$\kappa$-branch""<br />
through a tree $T$ is a maximal linearly ordered subset of<br />
$T$ of order type $\kappa$. Such a branch selects exactly<br />
one node from each level, in a linearly ordered manner. The set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree<br />
is an ''Aronszajn'' tree if it has no $\kappa$-branches.<br />
A cardinal $\kappa$ has the ''tree property'' if every<br />
$\kappa$-tree has a $\kappa$-branch.<br />
<br />
A transitive set $M$ is a $\kappa$-model of set theory if<br />
$|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$,<br />
the theory ZFC without the power set axiom &#40;and using collection and separation rather than merely replacement). <br />
For any<br />
infinite cardinal $\kappa$ we have<br />
that $H_{\kappa^+}$ models ZFC$^-$, and further, if<br />
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is<br />
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed<br />
into such an $M$. If $\kappa^{\lt\kappa}=\kappa$, one can use<br />
the downward L&ouml;wenheim-Skolem theorem to find such $M$<br />
with $M^{\lt\kappa}\subset M$. So in this case there are abundant<br />
$\kappa$-models of set theory &#40;and conversely, if there is<br />
a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$).<br />
<br />
The partition property $\kappa\to&#40;\lambda)^n_\gamma$<br />
asserts that for every function $F:[\kappa]^n\to\gamma$<br />
there is $H\subset\kappa$ with $|H|=\lambda$ such that<br />
$F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as<br />
coloring the $n$-tuples, the partition property asserts the<br />
existence of a ''monochromatic'' set $H$, since all<br />
tuples from $H$ get the same color. The partition property<br />
$\kappa\to&#40;\kappa)^2_2$ asserts that every partition of<br />
$[\kappa]^2$ into two sets admits a set $H\subset\kappa$ of<br />
size $\kappa$ such that $[H]^2$ lies on one side of the<br />
partition. When defining $F:[\kappa]^n\to\gamma$, we define<br />
$F&#40;\alpha_1,\ldots,\alpha_n)$ only when<br />
$\alpha_1<\cdots<\alpha_n$.<br />
<br />
== Weakly compact cardinals and the constructible universe ==<br />
<br />
Every weakly compact cardinal is weakly compact in [[Constructible universe|$L$]]. <cite>Jech2003:SetTheory</cite><br />
<br />
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. <br />
<br />
<br />
== Weakly compact cardinals and forcing ==<br />
<br />
* Weakly compact cardinals are invariant under small forcing. [http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf]<br />
* Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions {{Citation needed}}.<br />
* If $\kappa$ is weakly compact, there is a forcing extension in which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa^+$ {{Citation needed}}. <br />
* If the existence of weakly compact cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not weakly compact, but becomes weakly compact in a forcing extension <CITE>Kunen1978:SaturatedIdeals</CITE>.<br />
<br />
== Indestructibility of a weakly compact cardinal ==<br />
''To expand using [https://arxiv.org/abs/math/9907046]''<br />
<br />
== Relations with other large cardinals == <br />
<br />
* Every weakly compact cardinal is [[inaccessible]], [[Mahlo]], hyper-Mahlo, hyper-hyper-Mahlo and more. <br />
* [[Measurable]] cardinals, [[Ramsey]] cardinals, and [[indescribable|totally indescribable]] cardinals are all weakly compact and a stationary limit of weakly compact cardinals.<br />
* Assuming the consistency of a [[strongly unfoldable]] cardinal with ZFC, it is also consistent for the least weakly compact cardinal to be the least [[unfoldable]] cardinal. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If GCH holds, then the least weakly compact cardinal is not [[weakly measurable]]. However, if there is a [[measurable]] cardinal, then it is consistent for the least weakly compact cardinal to be weakly measurable. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If it is consistent for there to be a [[nearly supercompact]], then it is consistent for the least weakly compact cardinal to be nearly supercompact. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
* For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-[[Ramsey]]. <cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
==$\Sigma_n$-weakly compact etc.==<br />
An inaccessible cardinal $κ$ is $Σ_n$-weakly compact iff it reflects $Π_1^1$ sentences with $Σ_n$-predicates, i.e. for every $R ⊆ V_κ$ which is definable by a $Σ_n$ formula &#40;with parameters) over $V_κ$ and every $Π_1^1$ sentence $Φ$, if $\langle V_κ , ∈, R \rangle \models Φ$ then there is $α < κ$ &#40;equivalently, unboundedly-many $α < κ$) such that $\langle V_α , ∈, R ∩ V_α \rangle \models Φ$. Analogously for $Π_n$ and $∆_n$. $κ$ is $Σ_ω$-weakly compact iff it is $Σ_n$-weakly compact for all $n < ω$.<br />
<br />
$κ$ is $Σ_n$-weakly compact $\iff$ $κ$ is $Π_n$-weakly compact $\iff$ $κ$ is $∆_{n+1}$-weakly compact $\iff$ For every $Π_1^1$ formula $Φ&#40;x_0 , ..., x_k)$ in the language of set theory and every<br />
$a_0 , ..., a_k ∈ V_κ$, if $V κ \models Φ&#40;a_0 , ..., a_k )$, then there is $λ ∈ I_n := \{λ < κ : λ$ is inaccessible and $V_λ \preccurlyeq_n V_κ\}$ such that $V_λ \models Φ&#40;a_0 , ..., a_k)$.<br />
<br />
In <cite>Bosch2006:SmallDefinablyLargeCardinals</cite> it is shown that every $Σ_ω$-w.c. cardinal is $Σ_ω$-[[Mahlo]] and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.<br />
<br />
These properties are connected with [[axioms of generic absoluteness]]. For example:<br />
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.<br />
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.<br />
<br />
This section from<cite>Leshem2000:OCDefinableTreeProperty</cite><cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
==Recursive analogue==<br />
''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-[[admissible]]'' ordinals are analogous to weakly compact &#40;$Π_1^1$-indescribable) cardinals and can be called ''recursively weakly compact''<cite>Madore2017:OrdinalZoo</cite><cite>RichterAczel1974:InductiveDefinitions</cite><sup>after definition 1.12</sup><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Weakly_compact&diff=4162Weakly compact2022-05-15T06:07:35Z<p>BartekChom: 2-regular</p>
<hr />
<div>{{DISPLAYTITLE: Weakly compact cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
Weakly compact cardinals lie at the focal point of a number<br />
of diverse concepts in infinite combinatorics, admitting various characterizations in terms of these concepts. If $\kappa^{{<}\kappa} = \kappa$, then the following are equivalent: <br />
<br />
:; Weak compactness : A cardinal $\kappa$ is weakly compact if and only if it is [[uncountable]] and every $\kappa$-satisfiable theory in an [[Infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] language of size at most $\kappa$ is satisfiable.<br />
:; Extension property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset V_\kappa$, there is a transitive structure $W$ properly extending $V_\kappa$ and $A^*\subset W$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.<br />
:; Tree property : A cardinal $\kappa$ is weakly compact if and only if it is [[inaccessible]] and has the [[tree property]].<br />
:; Filter property : A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a set containing at most $\kappa$-many subsets of $\kappa$, then there is a $\kappa$-[[filter|complete nonprincipal filter]] $F$ measuring every set in $M$.<br />
:; Weak embedding property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with $\kappa\in M$ and a transitive set $N$ with an [[elementary embedding|embedding]] $j:M\to N$ with [[critical point]] $\kappa$.<br />
:; Embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a transitive set $N$ and an embedding $j:M\to N$ with critical point $\kappa$.<br />
:; Normal embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$, such that $N=\{\ j&#40;f)&#40;\kappa)\mid f\in M\ \}$.<br />
;; Hauser embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.<br />
:; Partition property : A cardinal $\kappa$ is weakly compact if and only if the [[partition property]] $\kappa\to&#40;\kappa)^2_2$ holds.<br />
:; Indescribability property : A cardinal $\kappa$ is weakly compact if and only if it is $\Pi_1^1$-[[indescribable]].<br />
:; Skolem Property : A cardinal $\kappa$ is weakly compact if and only if $\kappa$ is inaccessible and every $\kappa$-unboundedly satisfiable $\mathcal{L}_{\kappa,\kappa}$-theory $T$ of size at most $\kappa$ has a model of size at least $\kappa$. A theory $T$ is $\kappa$-unboundedly satisfiable if and only if for any $\lambda<\kappa$, there exists a model $\mathcal{M}\models T$ with $\lambda\leq|M|<\kappa$. For more info see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937#309937 here].<br />
:; 2-regular : A cardinal $\kappa$ is weakly compact if and only if every $\kappa$-bounded $F: \kappa_\kappa\rightarrow\kappa_\kappa$ has a witness &#40;$0<\alpha<\kappa$ such that for every $f: \kappa\rightarrow\kappa$ we have $f|\alpha\subseteq\alpha \imples F(f)"\alpha\subseteq\alpha$). ''TODO complete'' <cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.13<!--typo, written as 1.3-->, theorem 1.14</sup><br />
<br />
Weakly compact cardinals first arose<br />
in connection with &#40;and were named for) the question of<br />
whether certain [[Infinitary logic|infinitary logics]] satisfy the compactness<br />
theorem of first order logic. Specifically, in a language<br />
with a signature consisting, as in the first order context,<br />
of a set of constant, finitary function and relation<br />
symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$<br />
formulas by closing the collection of formulas under<br />
infinitary conjunctions<br />
$\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions<br />
$\vee_{\alpha<\delta}\varphi_\alpha$ of any size<br />
$\delta<\kappa$, as well as infinitary quantification<br />
$\exists\vec x$ and $\forall\vec x$ over blocks of<br />
variables $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less<br />
than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics.<br />
A theory is ''$\theta$-satisfiable'' if every subtheory<br />
consisting of fewer than $\theta$ many sentences of it is<br />
satisfiable. First order logic is precisely<br />
$L_{\omega,\omega}$, and the classical Compactness theorem<br />
asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$<br />
theory is satisfiable. A uncountable cardinal $\kappa$ is<br />
''[[strongly compact]]'' if every $\kappa$-satisfiable<br />
$\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal<br />
$\kappa$ is ''weakly compact'' if every<br />
$\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a<br />
language having at most $\kappa$ many constant, function<br />
and relation symbols, is satisfiable.<br />
<br />
Next, for any cardinal $\kappa$, a ''$\kappa$-tree'' is a<br />
tree of height $\kappa$, all of whose levels have size less<br />
than $\kappa$. More specifically, $T$ is a ''tree'' if<br />
$T$ is a partial order such that the predecessors of any<br />
node in $T$ are well ordered. The $\alpha^\textrm{th}$ level of a<br />
tree $T$, denoted $T_\alpha$, consists of the nodes whose<br />
predecessors have order type exactly $\alpha$, and these<br />
nodes are also said to have ''height'' $\alpha$. The height of the tree $T$ is the first $\alpha$ for which $T$<br />
has no nodes of height $\alpha$. A ""$\kappa$-branch""<br />
through a tree $T$ is a maximal linearly ordered subset of<br />
$T$ of order type $\kappa$. Such a branch selects exactly<br />
one node from each level, in a linearly ordered manner. The set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree<br />
is an ''Aronszajn'' tree if it has no $\kappa$-branches.<br />
A cardinal $\kappa$ has the ''tree property'' if every<br />
$\kappa$-tree has a $\kappa$-branch.<br />
<br />
A transitive set $M$ is a $\kappa$-model of set theory if<br />
$|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$,<br />
the theory ZFC without the power set axiom &#40;and using collection and separation rather than merely replacement). <br />
For any<br />
infinite cardinal $\kappa$ we have<br />
that $H_{\kappa^+}$ models ZFC$^-$, and further, if<br />
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is<br />
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed<br />
into such an $M$. If $\kappa^{\lt\kappa}=\kappa$, one can use<br />
the downward L&ouml;wenheim-Skolem theorem to find such $M$<br />
with $M^{\lt\kappa}\subset M$. So in this case there are abundant<br />
$\kappa$-models of set theory &#40;and conversely, if there is<br />
a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$).<br />
<br />
The partition property $\kappa\to&#40;\lambda)^n_\gamma$<br />
asserts that for every function $F:[\kappa]^n\to\gamma$<br />
there is $H\subset\kappa$ with $|H|=\lambda$ such that<br />
$F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as<br />
coloring the $n$-tuples, the partition property asserts the<br />
existence of a ''monochromatic'' set $H$, since all<br />
tuples from $H$ get the same color. The partition property<br />
$\kappa\to&#40;\kappa)^2_2$ asserts that every partition of<br />
$[\kappa]^2$ into two sets admits a set $H\subset\kappa$ of<br />
size $\kappa$ such that $[H]^2$ lies on one side of the<br />
partition. When defining $F:[\kappa]^n\to\gamma$, we define<br />
$F&#40;\alpha_1,\ldots,\alpha_n)$ only when<br />
$\alpha_1<\cdots<\alpha_n$.<br />
<br />
== Weakly compact cardinals and the constructible universe ==<br />
<br />
Every weakly compact cardinal is weakly compact in [[Constructible universe|$L$]]. <cite>Jech2003:SetTheory</cite><br />
<br />
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. <br />
<br />
<br />
== Weakly compact cardinals and forcing ==<br />
<br />
* Weakly compact cardinals are invariant under small forcing. [http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf]<br />
* Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions {{Citation needed}}.<br />
* If $\kappa$ is weakly compact, there is a forcing extension in which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa^+$ {{Citation needed}}. <br />
* If the existence of weakly compact cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not weakly compact, but becomes weakly compact in a forcing extension <CITE>Kunen1978:SaturatedIdeals</CITE>.<br />
<br />
== Indestructibility of a weakly compact cardinal ==<br />
''To expand using [https://arxiv.org/abs/math/9907046]''<br />
<br />
== Relations with other large cardinals == <br />
<br />
* Every weakly compact cardinal is [[inaccessible]], [[Mahlo]], hyper-Mahlo, hyper-hyper-Mahlo and more. <br />
* [[Measurable]] cardinals, [[Ramsey]] cardinals, and [[indescribable|totally indescribable]] cardinals are all weakly compact and a stationary limit of weakly compact cardinals.<br />
* Assuming the consistency of a [[strongly unfoldable]] cardinal with ZFC, it is also consistent for the least weakly compact cardinal to be the least [[unfoldable]] cardinal. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If GCH holds, then the least weakly compact cardinal is not [[weakly measurable]]. However, if there is a [[measurable]] cardinal, then it is consistent for the least weakly compact cardinal to be weakly measurable. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If it is consistent for there to be a [[nearly supercompact]], then it is consistent for the least weakly compact cardinal to be nearly supercompact. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
* For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-[[Ramsey]]. <cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
==$\Sigma_n$-weakly compact etc.==<br />
An inaccessible cardinal $κ$ is $Σ_n$-weakly compact iff it reflects $Π_1^1$ sentences with $Σ_n$-predicates, i.e. for every $R ⊆ V_κ$ which is definable by a $Σ_n$ formula &#40;with parameters) over $V_κ$ and every $Π_1^1$ sentence $Φ$, if $\langle V_κ , ∈, R \rangle \models Φ$ then there is $α < κ$ &#40;equivalently, unboundedly-many $α < κ$) such that $\langle V_α , ∈, R ∩ V_α \rangle \models Φ$. Analogously for $Π_n$ and $∆_n$. $κ$ is $Σ_ω$-weakly compact iff it is $Σ_n$-weakly compact for all $n < ω$.<br />
<br />
$κ$ is $Σ_n$-weakly compact $\iff$ $κ$ is $Π_n$-weakly compact $\iff$ $κ$ is $∆_{n+1}$-weakly compact $\iff$ For every $Π_1^1$ formula $Φ&#40;x_0 , ..., x_k)$ in the language of set theory and every<br />
$a_0 , ..., a_k ∈ V_κ$, if $V κ \models Φ&#40;a_0 , ..., a_k )$, then there is $λ ∈ I_n := \{λ < κ : λ$ is inaccessible and $V_λ \preccurlyeq_n V_κ\}$ such that $V_λ \models Φ&#40;a_0 , ..., a_k)$.<br />
<br />
In <cite>Bosch2006:SmallDefinablyLargeCardinals</cite> it is shown that every $Σ_ω$-w.c. cardinal is $Σ_ω$-[[Mahlo]] and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.<br />
<br />
These properties are connected with [[axioms of generic absoluteness]]. For example:<br />
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.<br />
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.<br />
<br />
This section from<cite>Leshem2000:OCDefinableTreeProperty</cite><cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
==Recursive analogue==<br />
''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-[[admissible]]'' ordinals are analogous to weakly compact &#40;$Π_1^1$-indescribable) cardinals and can be called ''recursively weakly compact''<cite>Madore2017:OrdinalZoo</cite><cite>RichterAczel1974:InductiveDefinitions</cite><sup>after definition 1.12</sup><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4161Reflecting ordinal2022-05-15T05:32:40Z<p>BartekChom: /* Properties */ theorem 1.9</p>
<hr />
<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.7</sup><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
==Properties==<br />
$Π_2$-reflecting ordinals are precisely the [[admissible]] ordinals $>\omega$ &#40;class $\mathrm{Ad}$). &#40;theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\alpha$ is a limit of $X$ &#40;$\alpha = \sup &#40;X \cap \alpha)$) $\iff$ $\alpha$ is $\Pi_0^0$-reflecting on $X$ $\iff$ $\alpha$ is $\Sigma_2^0$-reflecting on $X$. &#40;theorem 1.9 i)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
An ordinal is $\Pi_2^0$-reflecting on $X$ if it is recursively [[Mahlo]] on $X$. &#40;theorem 1.9 ii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
An ordinal is $\Pi_n^0$-reflecting on $X$ iff it is $\Sigma_{n+1}^0$-reflecting on $X$. &#40;theorem 1.9 iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
When $Q$ is $\Pi_m^n$ for $m>2$, $\Pi_m^n$ for $n>0$, $\Sigma_m^n$ for $m>3$ or $\Sigma_m^n$ for $n>0$, an ordinal is $Q$-reflecting on $X$ iff it is $Q$-reflecting on $X \cap \mathrm{Ad}$. &#40;theorem 1.9 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\alpha$ is $Q$-reflecting on $X$ iff $\alpha$ reflects every $Q$-sentence of $\mathcal{L}_p$ on $X$, where $\mathcal{L}_p$ is the sublanguage of $\mathcal{L}$ with relation<br />
symbols only for the primitive recursive relations on sets. ''TODO: complete'' &#40;theorem 1.10)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$Π_3$-reflecting ordinals are precisely the 2-[[admissible]] ordinals &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$&#40;+1)$-[[stable]] ordinals are exactly the $Π^1_0$-reflecting &#40;i.e., $Π_n$-reflecting for every $n ∈ ω$<cite>Madore2017:OrdinalZoo</cite>) ordinals &#40;Theorem 1.18). $&#40;{}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals &#40;Theorem 1.19).<cite>RichterAczel1974:InductiveDefinitions</cite><!--Page 18 in the PDF, with label 16--><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Forcing&diff=4160Forcing2022-05-14T20:27:20Z<p>BartekChom: /* Bounded Forcing Axiom */ BPFA implies $\mathfrak{c} = \aleph_2$</p>
<hr />
<div>'''Forcing''' is a method for extending a transitive model $M$ of $\text{ZFC}$ (the ''ground [[model]]'') by adjoining a new set $G$ (the ''generic set'') to produce a new, larger model $M[G]$ called a ''generic extension''. In short, the set $G$ can be constructed a certain way using a partially ordered set (poset) $(\mathbb{P},\leq)\in M$ (the ''forcing notion'') so that the following holds:<br />
<br />
* '''(Generic Model Theorem).''' There exists a unique transitive model $M[G]$ of $\text{ZFC}$ that includes $M$ (as a subset) and contains $G$ (as an element), has the same ordinals as $M$, and any transitive model of $\text{ZFC}$ also including $M$ and containing $G$ includes $M[G]$ (i.e. $M[G]$ is minimal).<br />
<br />
The elements of the forcing notion $\mathbb{P}$ will be called the ''conditions''. The order $p\leq q$, for $p,q\in\mathbb{P}$, is to be interpreted as "$p$ is stronger than $q$" or as "$p$ implies $q$". $G$ will be a special subset of $\mathbb{P}$ said to be ''generic over $M$'' and satisfying some requirements. The choice of $\mathbb{P}$ and of $\leq$ will determine what is true of false in $M[G]$. A special relation called the ''forcing relation'' is defined, which links the conditions to the formulas they will force. It is very important to note that this relation can be defined from within the ground model $M$.<br />
<br />
While the usual definition of forcing only works for transitive countable models $M$ of $\text{ZFC}$, it is customary to "take $V$ as the ground model", and pretend there exists a generic $G\subseteq\mathbb{P}$. Every statement about the generic extension $V[G]$ can be thought as a statement in the forcing relation: that relation being definable within the ground model, this method can be thought as working within the ground model $M$, with $V[G]$ being, in some way, $M[G]$ as seen from within the ground model $M$.<br />
<br />
Forcing was first introduced by Paul Cohen as a way of proving the consistency of the failure of the [[GCH|continuum hypothesis]] with $\text{ZFC}$. He also used it to prove the consistency of the failure of the [[axiom of choice]], albeit the proof is more indirect: if $M$ satisfies choice, then so does $M[G]$, so $\neg AC$ cannot be forced directly, though it is possible to extract a submodel of $M[G]$ (for a particular generic extension) in which choice fails.<br />
<br />
In particular, an inner model (a class-sized transitive model (of ZFC or a weaker theory) containing all ordinals) can be a ground of $V$.<br />
<br />
== Definitions and some properties ==<br />
<br />
Let $(\mathbb{P},\leq)$ be a partially ordered set, the ''forcing notion''. Sometimes $\leq$ can just be a preorder (i.e. not necessarily antisymmetric). The elements of $\mathbb{P}$ are called ''conditions''. We will assume $\mathbb{P}$ has a maximal element $1$, i.e. one has $p\leq 1$ for all $p\in\mathbb{P}$. This element isn't necessary if one uses the definition using Boolean algebras presented below, but is useful when trying to construct $M[G]$ without using Boolean algebras.<br />
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Two conditions $p,q\in P$ are ''compatible'' if there exists $r\in\mathbb{P}$ such that $r\leq p$ and $r\leq q$. They are ''incompatible'' otherwise. A set $W\subseteq\mathbb{P}$ is an ''antichain'' if all its elements are pairwise incompatible.<br />
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=== Genericity ===<br />
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A nonempty set $F\subseteq\mathbb{P}$ is a ''[[filter]] on $\mathbb{P}$'' if all of its elements are pairwise compatible and it is closed under implications, i.e. if $p\leq q$ and $p\in F$ then $q\in F$.<br />
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One says that a set $D\subseteq\mathbb{P}$ is ''dense'' if for all $p\in\mathbb{P}$, there is $q\in D$ such that $q\leq p$ (i.e. $q$ ''implies'' $p$). $D$ is ''open dense'' if additionally $q\leq p$ and $p\in D$ implies $q\in D$. $D$ is ''predense'' if every $p\in\mathbb{P}$ is compatible with some $q\in D$.<br />
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Given a collection $\mathcal{D}$ of dense subsets of $\mathbb{P}$, one says that a filter $G$ is '''$\mathcal{D}$-generic''' if it intersects all sets $D\in\mathcal{D}$, i.e. $D\cap G\neq\empty$.<br />
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Given a transitive model $M$ of $\text{ZFC}$ such that $(\mathbb{P},<)\in M$, we say that a filter $G\subseteq\mathbb{P}$ is '''$M$-generic''' (or $\mathbb{P}$-generic in $M$, or just generic) if it is $\mathcal{D}_M$-generic where $\mathcal{D}_M$ is the set of all dense subsets of $\mathbb{P}$ in $M$.<br />
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In the above definitions, ''dense'' can be replaced with ''open dense'', ''predence'' or ''a maximal antichain'', and the resulting notion of genericity would be the same.<br />
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In most cases, if $G$ is $\mathbb{P}$-generic over $M$ then $G\not\in M$. The Generic Model Theorem mentioned above says that there is a minimal model $M[G]\supseteq M$ with $M[G]\models\text{ZFC}$, $G\in M[G]$, and if $M\models$ "$x$ is an ordinal" then so does $M[G]$.<br />
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=== $\mathbb{P}$-names and interpretation by $G$ ===<br />
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Using transfinite recursion, define the following cumulative hierarchy:<br />
* $V^\mathbb{P}_0=\empty$, $V^\mathbb{P}_\lambda = \bigcup_{\alpha<\lambda}V^\mathbb{P}_\alpha$ for limit $\lambda$<br />
* $V^\mathbb{P}_{\alpha+1} = \mathcal{P}(V^\mathbb{P}_\alpha\times\mathbb{P})$<br />
* $V^\mathbb{P} = \bigcup_{\alpha\in\mathrm{Ord}}V^\mathbb{P}_\alpha$<br />
Elements of $V^\mathbb{P}$ are called ''$\mathbb{P}$-names''. Every nonempty $\mathbb{P}$-name is of a set of pairs $(n,p)$ where $n$ is another $\mathbb{P}$-name and $p\in\mathbb{P}$.<br />
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Given a filter $G\subseteq\mathbb{P}$, define the ''interpretation of $\mathbb{P}$-names'' by $G$: Given a $\mathbb{P}$-name $x$, let $x^G=\{y^G : ((\exists p\in G)(y,p)\in x)\}$. Letting $\breve{x}=\{(\breve{y},1):y\in x\}$ for every set $x$ be the ''canonical name'' for $x$, one has $\breve{x}^G=x$ for all $x$.<br />
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Let $M$ be a transitive model of $\text{ZFC}$ such that $(\mathbb{P},\leq)\subseteq M$. Let $M^\mathbb{P}$ be the $V^\mathbb{P}$ constructed in $M$. Given a $M$-generic filter $G\subseteq\mathbb{P}$, we can now define the generic extension $M[G]$ to be $\{x^G : x\in M^\mathbb{P}\}$. This $M[G]$ satisfies the Generic Model Theorem.<br />
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=== The forcing relation ===<br />
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Define the ''forcing language'' to be the first-order language of set theory augmented by a constant symbol for every $\mathbb{P}$-name in $M^\mathbb{P}$. Given a condition $p\in\mathbb{P}$, a formula $\varphi(x_1,...,x_n)$ and $x_1,...,x_n \in M^\mathbb{P}$, we say that '''$p$ forces $\varphi(x_1,...,x_n)$''', denoted $p\Vdash_ {M,\mathbb{P}}\varphi(x_1,...,x_n)$ if for all $M$-generic filter $G$ with $p\in G$ one has $M[G]\models\varphi(x_1^G,...,x_n^G)$. There exists an "internal" definition of $\Vdash$, i.e. a definition formalizable in $M$ itself, by induction on the complexity of the formulas of the forcing language.<br />
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The '''Forcing Theorem''' asserts that if $\sigma$ is a sentence of the forcing language, $M[G]$ satisfies $\sigma$ if and only if some condition $p\in G$ forces $\sigma$.<br />
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The forcing relation has the following properties, for all $p,q\in\mathbb{P}$ and formulas $\varphi,\psi$ of the forcing language:<br />
* $p\Vdash\varphi\land q\leq p\implies q\Vdash\varphi$<br />
* $p\Vdash\varphi\implies\neg(p\Vdash\neg\varphi)$<br />
* $p\Vdash\neg\varphi\iff\neg\exists q\leq p(q\Vdash\varphi)$<br />
* $p\Vdash(\varphi\land\psi)\iff(p\Vdash\varphi\land p\Vdash\psi)$<br />
* $p\Vdash\forall x\varphi\iff\forall x\in M^\mathbb{P}(p\Vdash\varphi(x))$<br />
* $p\Vdash(\varphi\lor\psi)\iff\forall q\leq p\exists r\leq q(r\Vdash\varphi\lor r\Vdash\psi)$<br />
* $p\Vdash\exists x\varphi\iff\forall q\leq p\exists r\leq q\exists x\in M^\mathbb{R}(r\Vdash\varphi(x))$<br />
* $p\Vdash\exists x\varphi\implies\exists x\in M^\mathbb{P}(p\Vdash\varphi(x))$<br />
* $\forall p\exists q\leq p (q\Vdash\varphi\lor q\Vdash\neg\varphi)$<br />
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=== Separativity ===<br />
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A forcing notion $(\mathbb{P},\leq)$ is ''separative'' if for all $p,q\in\mathbb{P}$, if $p\not\leq q$ then there exists a $r\leq p$ incompatible with $q$. Many notions aren't separative, for example if $\leq$ is a linear order than $(\mathbb{P},\leq)$ is separative iff $\mathbb{P}$ has only one element. However, every notion $(\mathbb{P},\leq)$ has a unique (up to isomorphism) ''separative quotient'' $(\mathbb{Q},\preceq)$, i.e. a notion $(\mathbb{Q},\preceq)$ and a function $i:\mathbb{P}\to\mathbb{Q}$ such that $x\leq y\implies i(x)\preceq i(y)$ and $x, y$ are compatible iff $i(x),i(y)$ are compatible. This name comes from the fact that $\mathbb{Q}=(\mathbb{P}/\equiv)$ where $x\equiv y$ iff every $z\in P$ is compatible with $x$ iff it is compatible with $y$. The order $\preceq$ on the equivalence classes is "$[x]\preceq[y]$ iff all $z\leq x$ are compatible with $y$". Also $i(x)=[x]$. It is sometimes convenient to identify a forcing notion with its separative quotient.<br />
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== Boolean algebras ==<br />
''To be expanded.''<br />
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It is sometimes convenient to take the forcing notion $\mathbb{P}$ to be a Boolean algebra $\mathbb{B}$.<br />
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== Consistency proofs ==<br />
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Let $T_1$ and $T_2$ be some recursively enumerable enumerable extensions of $\text{ZFC}$ (possibly $\text{ZFC}$ itself). The existence of a countable transitive model $M$ of the theory $T_1$ is equivalent to the assertion that $T_1$ is consistent. When we construct a generic extension $M[G]$ satisfying $T_2$ from a countable transitive model $M$ of $T_1$, we prove the consistency of $T_2$ (since we prove it has a set model) only from the consistency of $T_1$, i.e. we prove $\text{Con}(T_1)\implies\text{Con}(T_2)$.<br />
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For instance, by following Cohen's construction of a generic extension statisfying $\text{ZFC+}\neg\text{CH}$ from a model of $\text{ZFC}$, we prove that $\text{Con}(\text{ZFC})\implies\text{Con}(\text{ZFC+}\neg\text{CH})$. It follows that if $\text{ZFC}$ is consistent then it cannot prove $\text{CH}$, as otherwise $\text{ZFC+}\neg\text{CH}$ would be inconsistent, contradicting the above implication proved by forcing.<br />
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Those implications between the consistencies of different theories are the ''relative consistency results'' set theorists are often interested in. The subsection below provides many more examples of consistency results, where the theory $T_1$ above is often $\text{ZFC}$ augmented by large cardinal axioms. <br />
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=== Other examples of consistency results proved using forcing ===<br />
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In the following examples, the generated generic extensions satisfy the axiom of choice unless indicated otherwise.<br />
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* '''Easton's theorem:''' Let $M$ be a transitive set model of $\text{ZFC+GCH}$. Let $F$ be an increasing function in $M$ from the set of $M$'s regular cardinals to the set of $M$'s cardinals, such that for all regular $\kappa$, $\mathrm{cf}F(\kappa)>\kappa$. Then there is a generic extension $M[G]$ of $M$ with the same cardinals and cofinalities such that $M[G]\models\text{ZFC+}\forall\kappa($if $\kappa$ is regular then $2^\kappa=F(\kappa)$).<br />
* '''Violating the Singular Cardinal Hypothesis at $\aleph_\omega$:''' Assume there is a [[measurable]] cardinal of [[Mitchell order]] $o(\kappa)=\kappa^{++}$. Then there is a generic extension in which $\kappa=\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}$. The hypothesis used here is optimal: in term of consistency strength, no less than a measurable of order $\kappa^{++}$ can produce a model where $\text{SCH}$ fails.<br />
* '''Violating the Singular Cardinal Hypothesis everywhere:''' It is consistent relative to the existence of a $(\delta+2)-$[[strong]] cardinal $\delta$ that $2^\kappa=\kappa^+$ for every successor $\kappa$ but $2^\kappa=\kappa^{++}$ for every limit cardinal $\kappa$.<br />
* '''Violating the Generalized Continuum Hypothesis everywhere:''' It is consistent relative to the existence of a $(\delta+2)-$strong cardinal $\delta$ that $2^\kappa=\kappa^{++}$ for every $\kappa$, i.e. $\text{GCH}$ fails everywhere.<br />
* '''Large cardinal properties of $\aleph_1$:''' Let $\kappa$ be measurable/[[supercompact]]/[[huge]]. Then there is a (sub)model (of a generic extension) satisfying $\text{ZF(+}\neg\text{AC)}$ in which $\kappa=\aleph_1$ and $\omega_1$ is measurable/supercompact/huge (by the ultrafilter characterizations, not by the elementary embedding characterizations.)<br />
* '''Singularity of every uncountable cardinal:''' It is consistent relative to the existence of a proper class of [[strongly compact]] cardinals there is model of $\text{ZF}$ in which (the axiom of choice does not hold and) every uncountable cardinal is singular and has cofinality $\omega$. The existence of a such model also implies that the [[axiom of determinacy]] holds in the $L(\mathbb{R})$ of some forcing extension of $\text{HOD}$.<br />
* '''[[Projective|Regularity properties]] of all sets of reals:''' Assume there is an [[inaccessible]] cardinal $\kappa$. Then there is a (sub)model (of a generic extension) that satisfies $\text{ZF+DC+}\neg\text{AC}$ and in which $\kappa=2^{\aleph_0}$ and every set of reals is Lebesgue measurable, has the Baire property and the perfect subset property. There is also a generic extension in which choice holds and every [[projective]] set of reals has those properties.<br />
* '''Real-valued measurability of the continuum:''' Assume there is a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$ and $2^{\aleph_0}$ is real-valued measurable (and thus weakly inaccessible, weakly hyper-[[Mahlo]], etc.)<br />
* '''Precipitousness of the [[filter|nonstationary ideal]] on $\omega_1$:''' Assume there is a measurable cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_1$ and the nonstationary ideal on $\omega_1$ is precipitous.<br />
* '''Saturation of the nonstationary ideal on $\omega_1$:''' Assume there is a [[Woodin]] cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_2$ the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated.<br />
* '''Saturation of an ideal on the continuum:''' Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$, there is a $2^{\aleph_0}$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ and there isn't any $\lambda$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ for every infinite $\lambda<2^{\aleph_0}$.<br />
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Some other applications of forcing:<br />
<!--* There is a generic extension in which there is a cardinal $\kappa$ such that $2^{\mathrm{cf}(\kappa)}<\kappa<\kappa^+<\kappa^{\mathrm{cf}(\kappa)}<2^\kappa$.--><br />
* It is consistent relative to the existence of an inaccessible cardinal that there are no Kurepa trees.<br />
* Let $\kappa$ be a [[superstrong]] cardinal. Let $V[G]$ be the generic extension of $V$ by the Lévy collapse $\mathrm{Coll}(\aleph_0,<\kappa)$. Then there is a nontrivial [[elementary embedding]] $j:L(\mathbb{R})\to(L(\mathbb{R}))^{V[G]}$.<br />
* Let $\kappa$ be a superstrong cardinal. There exists a $\omega$-distributive $\kappa$-c.c. notion of forcing $(\mathbb{P},\leq)$ such that in $V^\mathbb{P}$, $\kappa=\aleph_2$ and there exists a normal $\omega_2$-saturated $\sigma$-complete ideal on $\omega_1$.<br />
* Let $\kappa$ be a [[weakly compact]] cardinal. Then there is a generic extension in which $\kappa=\aleph_2$ and $\omega_2$ has the tree property. In fact, if there is infinitely many weakly compact cardinals then in a generic extension $\omega_{2n}$ has the tree property for every $n$. [http://logika.ff.cuni.cz/radek/papers/Friedman_Honzik_treeprop_revised.pdf]<br />
* It is consistent relative to the existence of infinitely many supercompact cardinals that there exists infinitely many cardinals $\delta$ above $2^{\aleph_0}$ such that both $\delta$ and $\delta^+$ have the tree property. Also, the [[axiom of projective determinacy]] holds in any such model.<br />
* Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa$ remains weakly compact, there is a $\kappa^+$-saturated $\kappa$-complete ideal on $\kappa$ but there isn't any $\kappa$-saturated $\kappa$-complete ideal on $\kappa$. One can replace "$\kappa$ is weakly compact" by "$\kappa$ is weakly inaccessible and $\kappa=2^{\aleph_0}$".<br />
* It is consistent relative to a supercompact cardinal that there is an inaccessible cardinal $\kappa$, a cardinal $\lambda>\kappa$ and a stationary set $S\subseteq\mathcal{P}_\kappa(\lambda)$ that cannot be partitioned into $\kappa^+$ disjoint stationary subsets.<br />
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== Types of forcing ==<br />
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=== Chain conditions, distributivity, closure and property (K) ===<br />
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A forcing notion $(\mathbb{P},\leq)$ satisfies the ''$\kappa$-chain condition'' ($\kappa$-c.c.) if every antichain of elements of $\mathbb{P}$ has cardinality less than $\kappa$. The $\omega_1$-c.c. is called the ''countable chain condition'' (c.c.c.). An important feature of chain conditions is that if $(\mathbb{P},\leq)$ satisfies the $\kappa$-c.c. then if $\kappa$ is regular in $M$ then it will be regular in $M[G]$. Since the $\kappa$-c.c. implies the $\lambda$-c.c. for all $\lambda\geq\kappa$, it follows that the $\kappa$-c.c. implies all regular cardinals $\geq|\mathbb{P}|^+$ will be preserved, and in particular the c.c.c. implies all cardinals and cofinalities of $M$ will be preserved in $M[G]$ for all $M$-generic $G\subseteq\mathbb{P}$.<br />
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Let $\kappa$ be a regular uncountable cardinal. If $(\mathbb{P},\leq)$ is a $\kappa$-c.c. notion of forcing then every club $C\in M[G]$ of $\kappa$ has a subset $D$ that is a club subset of $\kappa$ in the ground model; therefore every stationary subset of $\kappa$ remains stationary in $M[G]$.<br />
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$(\mathbb{P},\leq)$ is ''$\kappa$-distributive'' if the intersection of $\kappa$ open dense sets is still open dense. $\kappa$-distributive notions for infinite $\kappa$ does not add new subsets to $\kappa$. A stronger property, closure, is defined the following way: $\mathbb{P}$ is ''$\kappa$-closed'' if every $\lambda\leq\kappa$, every descending sequence $p_0\geq p_1\geq...\geq p_\alpha\geq... (\alpha<\lambda)$ has a lower bound. Every $\kappa$-closed notion is $\kappa$-distributive. If, for some regular uncountable cardinal $\kappa$ and all $\lambda<\kappa$, $(\mathbb{P},\leq)$ is a $\lambda$-closed forcing notion, then every stationary subset of $\kappa$ remains stationary in every generic extension.<br />
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$(\mathbb{P},\leq)$ has ''property (K)'' (or just ''Knaster property'') if every uncountable set of conditions has an uncountable subet of pairwise compatible elements. Every notion with property (K) satisfies the c.c.c.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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A poset is ''productive-ccc'', if its product with any ccc poset is also ccc (in short $\textit{Prod-ccc}$).<br />
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A ccc poset $\mathbb{P}$ is ''strongly-$\underset{\sim}{Σ_n}$'' if it is $Σ_n$-definable in $H(ω_1)$ with parameters, and the predicate “$x$ codes a maximal antichain of $\mathbb{P}$” is also $Σ_n$-definable in $H(ω_1)$ with parameters.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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A [[projective]] poset $\mathbb{P}$ is ''absolutely-ccc'' if it is ccc in every inner model $W$ of $V$ which satisfies ZFC and contains the parameters of the definition of $\mathbb{P}$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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=== Cohen forcing, adding subsets of regular cardinals, and independence of the continuum hypothesis ===<br />
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Let $\kappa$ be a regular cardinal satisfying $2^{<\kappa}=\kappa$. Let $\lambda>\kappa$ be a cardinal such that $\lambda^\kappa=\lambda$. Let $\text{Add}(\kappa,\lambda) = (\mathbb{P},\leq)$ be the following partial order: $\mathbb{P}$ is the set of all functions $p$ with $\text{dom}(p)\subseteq\lambda\times\kappa$, $|\text{dom}(p)|<\kappa$ and $\text{ran}(p)\subseteq\{0,1\}$, and let $p\leq q$ iff $p\supseteq q$. Let $G$ be a $V$-generic on $\mathbb{P}$ and let $f=\bigcup G$. Then in $V[G]$, $f$ is a function from $\lambda\times\kappa$ to $\{0,1\}$. For every particular $\alpha<\lambda$, the function $c_\alpha(\xi)=f(\alpha,\xi)$ is in $V[G]$ the characteristic function of a subset $x_\alpha=\{\xi<\kappa:c_\alpha(\xi)=1\}$ of $\kappa$. None of those new subsets were originally in $V$, and if $\alpha\neq\beta$ then $x_\alpha\neq x_\beta$. Then, because $\mathbb{P}$ satisfies the $\kappa^+$-chain condition, it follows that all cardinals are preserved except that $2^\kappa=\lambda$.<br />
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In the special case $\kappa=\aleph_0$, there are new real numbers in $V[G]$ and $2^{\aleph_0}=\lambda$. Those new real numbers are called ''Cohen reals''. This technique allows one to show that $\text{ZFC}$ is consistent with the negation of the continuum hypothesis, i.e. that $2^{\aleph_0}>\aleph_1$. In fact, $2^{\aleph_0}$ can be any cardinal with uncountable cofinality, even if singular, e.g. one can force $2^{\aleph_0}=\aleph_{\omega_1}$. Note that $2^{\aleph_0}$ cannot be a cardinal of countable cofinality, so this is impossible to force.<br />
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=== Axiom A ===<br />
A forcing notion $(\mathbb{P},\leq)$ satisfies ''Axiom A'' if there is a sequence of partial orderings $\{\leq_n:n<\omega\}$ of $\mathbb{P}$ such that $p\leq_0 q$ implies $p\leq q$, for all n $p\leq_{n+1} q$ implies $p\leq_n q$, and the following conditions holds:<br />
* for every descending sequence $p_0\geq_0 p_1\geq_1...\geq_{n-1}p_n\geq_n...$ there is a $q$ such that $q\leq_n p_n$ for all $n$.<br />
* for every $p\in\mathbb{P}$, for every $n$ and every ordinal name $\alpha$ there exists $q\leq_n p$ and a countable set $B$ such that $p\Vdash\alpha\in B$.<br />
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Every c.c.c. or $\omega$-closed notion satisfies Axiom A.<br />
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=== Proper forcing ===<br />
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We say that a forcing notion $(\mathbb{P},\leq)$ is ''proper'' if for every uncountable cardinal $\lambda$, every [[stationary]] subset of $[\lambda]^\omega$ remains stationary in every generic extension.<br />
* Every c.c.c. or $\omega$-closed notion is proper, and so is every notion satisfying Axiom A.<br />
* Proper forcing does not collapse $\omega_1$: if $\mathbb{P}$ is proper then every countable set of ordinals in $M[G]$ is a subset of a countable set in $M$.<br />
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''Semi-proper forcing'' ''TODO''<br />
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A projective poset $\mathbb{P}$ is ''strongly-proper'' if for every countable transitive model $N$ of a fragment of ZFC with the parameters of the definition of $\mathbb{P}$ in $N$ and such that $(\mathbb{P}^N, ≤_\mathbb{P}^N, ⊥_\mathbb{P}) ⊆ (\mathbb{P}, ≤_\mathbb{P}, ⊥_\mathbb{P})$, and for every $p ∈ \mathbb{P}^N$, there is $q ≤ p$ which is $(N, \mathbb{P})$-generic, i.e., if $N \models$ “$A$ is a maximal antichain of $\mathbb{P}$”, then $A ∩ N$ is predense below $q$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
* A projective strongly-proper poset is proper.<br />
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=== The Lévy collapse ===<br />
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=== Prikry forcing ===<br />
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=== Other types of forcing, relations ===<br />
(subsection from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>)<br />
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Among important classes of posets are:<br />
* ''$σ$-centered'' posets (can be partitioned into countably many classes so that each class is finite-wise compatible)<br />
* ''$σ$-linked'' posets (can be partitioned into countably many classes so that each class is pair-wise compatible)<br />
* posets that preserve stationary subsets of $ω_1$ (in short $\textit{Stat-pres}$)<br />
* posets that preserve $ω_1$ (in short $\textit{$ω_1$-pres}$)<br />
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We have<br />
: $\textit{$\sigma$-centered} \subset \textit{$\sigma$-linked} \subset \textit{Knaster} \subset \textit{Prod-ccc} \subset \textit{ccc} \subset \textit{Proper} \subset \textit{Semi-proper} \subset \textit{Stat-pres} \subset \textit{$ω_1$-pres}$<br />
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Besides:<br />
* the poset for adding a Cohen real ($\textit{Cohen}$, compare subsection above)<br />
* the poset for adding a random real ($\textit{Random}$)<br />
* the amoeba poset for measure ($\textit{Amoeba}$)<br />
* the amoeba poset for category ($\textit{Amoeba-category}$)<br />
* the Hechler forcing for adding a dominating real<br />
* the Mathias forcing<br />
* Borel forcing notions (the set of conditions is a Borel set)<br />
* the $σ$-linked forcing notion for adding $ω_1$ random reals ($\textit{$ω_1$-Random}$)<br />
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== Product forcing ==<br />
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== Iterated forcing ==<br />
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== Forcing axioms ==<br />
=== Martin's axiom ===<br />
'''Martin's axiom''' ($\text{MA}$) is the following assertion: If $(\mathbb{P},\leq)$ is a forcing notion that satisfies the countable chain condition and if $\mathcal{D}$ is a collection of fewer than $2^{\aleph_0}$ dense subsets of $\mathbb{P}$, then there exists a $\mathcal{D}$-generic filter on $\mathbb{P}$. By replacing "fewer than $2^{\aleph_0}$" by "at most $\kappa$" on obtain the axiom $\text{MA}_\kappa$. Martin's axiom is then $\text{MA}_{<2^{\aleph_0}}$. Note that $\text{MA}_{\aleph_0}$ is provably true in $\text{ZFC}$.<br />
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For all $\kappa$, $\text{MA}_\kappa$ implies $\kappa<2^{\aleph_0}$. Martin's axiom follows from the continuum hypothesis, but is also consistent with its negation. $\text{MA}_{\aleph_1}$ implies there are no [[Suslin tree|Suslin trees]], that every [[tree property|Aronszajn tree]] is special, and that the c.c.c. is equivalent to property (K).<br />
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Martin's axiom implies that $2^{\aleph_0}$ is regular, that it is not real-valued measurable, and also that $2^\lambda=2^{\aleph_0}$ for all $\lambda<2^{\aleph_0}$. It implies that the intersection of fewer than $2^{\aleph_0}$ dense open sets is dense, the union of fewer than $2^{\aleph_0}$ null sets is null, and the union of fewer than $2^{\aleph_0}$ meager sets is meager. Also, the Lebesgue measure is $2^{\aleph_0}$-additive. If additionally $\neg\text{CH}$ then every $\mathbf{\Sigma}^1_2$ set is Lebesgue measurable and has the Baire property.<br />
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=== Proper Forcing Axiom ===<br />
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The '''Proper Forcing Axiom''' ($\text{PFA}$) is obtained by replacing "c.c.c." by "proper" in the definition of $\text{MA}_{\aleph_1}$: for every proper forcing notion $(\mathbb{P},\leq)$, if $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. $\text{PFA}$ implies $2^{\aleph_0}=\aleph_2$ and that the continuum (i.e. $\aleph_2$) has the [[tree property]]. It also implies that every two $\aleph_1$-dense sets of reals are isomorphic.<br />
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Unlike Martin's axiom, which is equiconsistent with $\text{ZFC}$, the $\text{PFA}$ has very high consistency strength, slightly below that of a [[supercompact]] cardinal. If there is a supercompact cardinal then there is a generic extension in which that supercompact is $\aleph_2$ and $\text{PFA}$ holds. On the other hand, [http://www.math.uni-bonn.de/people/pholy/acc_accepted.pdf] proves a ''quasi lower bound'' on the consistency strength of the $\text{PFA}$, which is at least the existence of a proper class of [[subcompact]] cardinals. [https://ac.els-cdn.com/S0001870811002635/1-s2.0-S0001870811002635-main.pdf?_tid=86c2030e-cca4-11e7-b23b-00000aab0f26&acdnat=1511039455_137e37101cda34d46bb0f195cbe43148] also shows that all known methods of forcing $\text{PFA}$ requires a [[strongly compact]] cardinal, and if one wants the forcing to be proper, a supercompact is required.<br />
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$\text{PFA}$ implies the failure of the [[square principle]] $\Box_\kappa$ for every uncountable cardinal $\kappa$, therefore it implies the [[axiom of determinacy|axiom of quasi-projective determinacy]]. It also implies the '''Open Coloring Axiom:''' let $X$ be a set of reals, and let $K\subseteq[X]^2$. We say that $K$ is ''open'' if the set $\{(x,y):\{x,y\}\in K\}$ is an open set in the space $X\times X$. Then<br />
* '''($\text{OCA}$).''' For every $X\subseteq\mathbb{R}$, and for any partition $[X]^2=K_0\cup K_1$ with $K_0$ open, either there exists an uncountable $Y\subseteq X$ such that $[Y]^2\subseteq K_0$ or there exists sets $H_n, n<\omega$ such that $X=\bigcup_{n<\omega}H_n$ and $[H_n]^2\subseteq K_1$ for all $n$.<br />
This axiom has many useful implications in combinatorial set theory.<br />
<br />
Statement equivalent to $\text{PFA}$: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.<cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite><br />
<br />
=== Martin's maximum and the semiproper forcing axiom ===<br />
<br />
'''Martin's Maximum''' is a strengthening of the proper forcing axiom defined the following way: suppose $(\mathbb{P},\leq)$ is a forcing notion that preserves stationary subsets of $\omega_1$, and that $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. This implies the proper forcing axiom, and also that the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated. It also implies that for all $\kappa\geq\aleph_2$, if $\kappa$ is regular then $\kappa^{\aleph_0}=\kappa$.<br />
<br />
=== Bounded Forcing Axiom ===<br />
&#40;Section from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>)<br />
<br />
The '''Bounded Forcing Axiom''' for a partial ordering $\mathbb{P}$, in short $BFA&#40;\mathbb{P})$, is the statement:<br />
: For every collection $\{I_α : α < ω_1 \}$ of maximal antichains of $\mathbf{B} \overset{def}{=} r.o.&#40;\mathbb{P}) \setminus \{\mathbf{0}\}$, each of size $≤ ω_1$, there is a filter $G ⊆ \mathbf{B}$ such that for every $α$, $I_α ∩ G ≠ ∅$.<br />
For a class of posets $Γ$, $BFA&#40;Γ)$ is the statement that for every $\mathbb{P} ∈ Γ$, $BFA&#40;\mathbb{P})$.<br />
<br />
Examples:<br />
* $MA_{ω_1}$, Martin’s axiom for $ω_1$, is $BFA&#40;\textit{ccc})$.<br />
* $BPFA$, the bounded proper forcing axiom, is $BFA&#40;\textit{Proper})$.<br />
* $BSPFA$, the bounded semi-proper forcing axiom, is $BFA&#40;\textit{Semi-proper})$.<br />
* $BMM$, the bounded Martin’s maximum, is $BFA&#40;\textit{Stat-pres})$.<br />
<br />
$BPFA$ implies that there is a well-ordering of the reals in length $ω_2$ definable with parameters in $H&#40;ω_2)$ and therefore $\mathfrak{c} = \aleph_2$.<br />
<br />
$BMM$ implies that for every set $X$ there is an inner model with a [[strong]] cardinal containing $X$.<br />
* Thus, in particular, $BMM$ implies that for every set $X$, [[zero dagger|$X^\dagger$ exists]].<br />
<br />
=== $\text{wPFA}$ and $\text{PFA}_κ$ ===<br />
(information in this subsection from <cite>BagariaGitmanSchindler2017:VopenkaPrinciple</cite>)<br />
<br />
The '''weak Proper Forcing Axiom''' is obtained by requiring only that embedding $j$ (like in the last statement equivalent to $\text{PFA}$) exists in a forcing extension: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ in $V$ and an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ in a set-forcing extension (equivalently in $V^{Coll(ω, \bar{M})}$) such that $φ(\bar{\mathcal{M}})$ holds.<br />
<br />
If there is a [[remarkable]] cardinal, then $\text{wPFA}$ holds in a forcing extension by a proper poset. If $\text{wPFA}$ holds, then $ω_2^V$ is remarkable in $L$.<br />
<br />
For a cardinal $κ$, $\text{PFA}_κ$ is the statement that<br />
: if $\mathbb{B}$ is any proper complete Boolean algebra and if $\langle A_ξ | ξ < ω_1 \rangle$ is any family of maximal antichains in $\mathbb{B}$ with $|A_ξ| ≤ κ$ for each $ξ < ω_1$, then there is some filter $G ⊆ \mathbb{B}$ such that $\forall_{ξ < ω_1} G ∩ A_ξ ≠ ∅$.<br />
Equivalently, in analogy to the other statements (adding the assumption $|M| ≤ κ$):<br />
: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $|M| ≤ κ$, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.<br />
<br />
$\text{PFA}_{\aleph_1}$ is $\text{BPFA}$.<br />
<br />
$\text{wPFA}$ implies $\text{PFA}_{\aleph_2}$. However, it does not imply $\text{PFA}_{\aleph_3}$, because assertion $\text{wPFA} ∧ ∀_{κ ≥ \aleph_2} \square_κ$ is consistent relative to a remarkable cardinal and (Todorcevic, 1984, Theorem 1) $\text{PFA}_{\aleph_3}$ implies the failure of $\square_{ω_2}$.<br />
<br />
== Generic absoluteness and universal Baireness ==<br />
<br />
''Main article: [[Universally Baire]]''<br />
<br />
== Generic ultrapowers ==<br />
<br />
''Main article: [[Filter#Precipitous ideals|Precipitous ideals]]''<br />
<br />
== Axioms of generic absoluteness ==<br />
<br />
''Main article: [[Axioms of generic absoluteness]]''<br />
<br />
{{stub}}<br />
<br />
{{references}}<br />
<br />
[[Category:Forcing]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Axioms_of_generic_absoluteness&diff=4159Axioms of generic absoluteness2022-05-14T20:15:34Z<p>BartekChom: /* Results for $H&#40;ω_2)$ and $Σ_1$ */ correction, +2</p>
<hr />
<div>&#40;from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>; compare [[Projective#Generically_absolute]])<br />
<br />
'''Axioms of generic absoluteness''' are axioms $\mathcal{A}&#40;W, \Phi, \Gamma)$ of the form “$W$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ \Gamma$”, where<br />
* $W$ is a subclass of $V$.<br />
* $\Phi$ is a class of sentences.<br />
* $\Gamma$ is a class of [[forcing]] notions.<br />
* “$W^V$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$” &#40;symbolically $W^V \equiv_\Phi W^{V^\mathbb{P}}$) means that $\forall_{\phi\in\Phi} &#40;W^V \models \phi \quad \text{iff} \quad W^{V^\mathbb{P}} \models \phi)$.<br />
$W$$, \Phi$ and $\Gamma$ must be definable classes for $\mathcal{A}&#40;W, \Phi, \Gamma)$ to be a sentence in the first-order language of Set Theory.<br />
<br />
== Notation ==<br />
* If $Γ$ contains only one element, $\mathbb{P}$, then one can write $\mathcal{A}&#40;W, Φ, \mathbb{P})$ instead of $\mathcal{A}&#40;W, Φ, Γ)$.<br />
* If $Γ$ is the class of all set-forcing notions, then one can just write $\mathcal{A}&#40;W, Φ)$.<br />
* The class of $\Sigma_n$ sentences with parameters from $W$ is denoted $\Sigma_n&#40;W)$ or in short $\underset{\sim}{\Sigma_n}$.<br />
** Analogously for $\Pi_n$ etc.<br />
** Boldface $\mathbf{\Sigma_n}$ is used in other sources for similar notions.<br />
<br />
== Basic properties ==<br />
* If $Φ ⊆ Φ_0$ and $Γ ⊆ Γ_0$, then $\mathcal{A}&#40;W, Φ_0 , Γ_0)$ implies $\mathcal{A}&#40;W, Φ, Γ)$.<br />
* $\mathcal{A}&#40;W, Φ, Γ)$ is equivalent to $\mathcal{A}&#40;W, \bar{Φ}, Γ)$, where $\bar{Φ}$ is the closure of $Φ$ under finite Boolean combinations.<br />
** Eg. $\mathcal{A}&#40;W, Σ_n , Γ)$ is equivalent to $\mathcal{A}&#40;W, Π_n , Γ)$<br />
* If $Φ ⊆ \underset{\sim}{Σ_0}$, then $\mathcal{A}&#40;W, Φ, Γ)$ holds for all transitive $W$ and all $Γ$ such that $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.<br />
* If $Φ ⊆ Σ_1&#40;H&#40;ω_1))$, then &#40;by the Levy-Shoenfield absoluteness theorem) $\mathcal{A}&#40;W, Φ, Γ)$ holds for every transitive model $W$ of a weak fragment of ZF that contains the parameters of $Φ$, and all $Γ$, provided $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.<br />
** In particular, the following hold:<br />
*** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_1})$<br />
*** $\mathcal{A}&#40;H&#40;κ), Σ_1&#40;H&#40;ω_1)))$ for $κ > ω_1$<br />
*** $\mathcal{A}&#40;V, Σ_1&#40;H&#40;ω_1)))$<br />
* $\mathcal{A}&#40;V, \underset{\sim}{Σ_1}, \mathbb{P})$ fails for any nontrivial $\mathbb{P}$ &#40;any $\mathbb{P}$ that adds some new set), because a proper class $W \neq V$ can never be an elementary substructure of $V$, since otherwise, by elementarity, $V_α^W = V_α$ for every ordinal $α$ and so $W = V$ &#40;contradiction).<br />
* For every forcing notion $\mathbb{P}$, $L^V = L^{V^\mathbb{P}}$, so $A&#40;L, Φ)$ holds for all $Φ$.<br />
* For every forcing notion $\mathbb{P}$, $H&#40;ω)^V = H&#40;ω)^{V^\mathbb{P}}$, so $A&#40;H&#40;ω), Φ)$ holds for all $Φ$.<br />
<br />
We see that, when $W = V$, $\Phi$ is the class of all sentences or $\Gamma$ is the class of all forcing notions, then the other two must be very small for the axiom to be consistent with ZFC.<br />
<br />
== Results ==<br />
Interesting results are obtained for $W = H&#40;κ)$ or $W = L&#40;H&#40;κ))$ with some definable uncountable cardinal $κ$.<br />
* $H&#40;κ)$ is better then $V_\alpha$ &#40;for ordinal $\alpha$), because<br />
** for regular $κ$ it is a model of ZFC without powerset and so it satisfies replacement.<br />
*** This allows for nice properties like: if $\mathbb{P} ∈ H&#40;κ)$, then a filter $G ⊆ \mathbb{P}$ is generic over $V$ iff it is generic over $H&#40;κ)$.<br />
** If $κ < λ$, then $\mathcal{A}&#40;H&#40;λ), \underset{\sim}{Σ_1}, Γ)$ implies $\mathcal{A}&#40;H&#40;κ), \underset{\sim}{Σ_1}, Γ)$.<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_2$ ===<br />
Relations with large cardinal properties:<br />
* If [[zero sharp|$X^\sharp$]] &#40;[[zero sharp|sharp]]) exists for every set $X$, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ holds.<br />
* The following are equiconsistent with ZFC:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_2)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ does not imply that $ω_1^L$ is countable.<br />
* If $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ holds after forcing with a certain proper poset, then either $ω_1$ is [[Mahlo]] in $L$ or $ω_2$ is [[inaccessible]] in $L$.<br />
* The following are equiconsistent with the existence of a $Σ_2$-[[reflecting cardinals|reflecting]] cardinal:<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$<br />
*** &#40;Because $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$ implies that $ω_1$ is a $Σ_2$-reflecting cardinal in $L[x]$ for every real $x$.)<br />
<br />
Relations with bounded forcing axioms:<br />
* $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$.<br />
* $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$.<br />
* $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.<br />
* $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.<br />
* The last four implications cannot be reversed, because all axioms of the form $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Stat-pres})$ are preserved after collapsing the continuum to $ω_1$ by $σ$-closed forcing and so are all consistent with CH and do not imply any of the bounded forcing axioms.<br />
* If $θ$ is the statement that every subset of $ω_1$ is constructible from a real, that is, for every $X ⊆ ω_1$ there is $x ⊆ ω$ with $X ∈ L[x]$ and<br />
** $ω_1$ is not [[weakly compact]] in $L[x]$ for some $x ⊆ ω$, then:<br />
** $MA_{ω_1}$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$ plus $θ$.<br />
** $BPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ plus $θ$.<br />
** $BSPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ plus $θ$.<br />
* $BSPFA$ is consistent with $ω_1^L = ω_1$.<br />
* $BMM$ implies that $ω_1$ is weakly compact in $L[x]$ for every $x ⊆ ω$.<br />
<br />
Equivalences to other statements:<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ and $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba-category})$ are both equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals has the property of Baire.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Lebesgue measurable.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Mathias})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Ramsey.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ is equivalent to the statement that every $\underset{\sim}{\Delta^1_2}$ set of reals is [[Projective#Suslin_sets_and_universally_Baire_sets|universally Baire]].<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_3$ ===<br />
Relations with large cardinal properties:<br />
* Each of the following implies that $ω_1$ is inaccessible in $L[x]$ for every real $x$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Cohen})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Hechler})$<br />
* The following are equiconsistent with the existence of a sharp for each set:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Stat-pres})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{$ω_1$-pres})$ &#40;obviously from the other two)<br />
* The existence of a $Σ_2$-reflecting cardinal and a sharp for each set is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$.<br />
* The following are equiconsistent with the existence of a weakly compact cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Knaster})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{ccc})$<br />
* The following are equiconsistent with the existence of a Mahlo cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-centered})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-linked})$<br />
* The following are equiconsistent with the existence of an inaccessible cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of posets that are absolutely-ccc and strongly-$\underset{\sim}{Σ_2}$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of strongly-proper posets that are $Σ_2$ definable in $H&#40;ω_1)$ with parameters<br />
** &#40;This result is optimal, for there is a, provably in ZFC, ccc poset $\mathbb{P}$ which is both $Σ_2$ and $Π_2$ definable in $H&#40;ω_1)$, without parameters, and for which the axiom $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \mathbb{P})$ fails if $ω_1$ is not a $Π_1$-Mahlo cardinal in $L$.)<br />
* The following are equiconsistent with the existence of a [[remarkable]] cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Proper})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Proper})$<br />
<br />
Relations with bounded forcing axioms:<br />
* If $x^\sharp$ exists for every real $x$ and the second uniform indiscernible is $< ω_2$, then<br />
** $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{ccc})$.<br />
** $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Proper})$.<br />
** $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Semi-proper})$.<br />
** $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Stat-pres})$.<br />
<br />
Relations with other statements:<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba-category}, \textit{Cohen}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals has the property of Baire.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba}, \textit{Random}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is Lebesgue measurable.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$ implies that every $\underset{\sim}{\Sigma^1_3}$ set of reals is Ramsey.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is universally Baire.<br />
** The converse does not hold.<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_n$, $4 \le n \le ω$ ===<br />
* $\mathcal{A}&#40;H&#40;ω_1), Σ_4)$ implies that [[zero dagger|$X^\dagger$ &#40;dagger)]] exists for every set $X$.<br />
* The following are equiconsistent with the existence of infinitely many [[strong]] cardinals:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_ω)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$<br />
* If there is a proper class of [[Woodin]] cardinals, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$.<br />
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.<br />
* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.<br />
* The consistency strength of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n})$ for $n \ge 4$ is<br />
** at least that of $n-3$ strong cardinals<br />
** and at most that of $n-3$ strong cardinals with a $Σ_2$-reflecting cardinal above them.<br />
<br />
=== Results for $H&#40;ω_2)$ and $Σ_1$ ===<br />
Relations with large cardinal properties:<br />
* The following are equiconsistent with the existence of a $Σ_2$-reflecting cardinal:<br />
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper})$<br />
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper})$<br />
<br />
Equivalence to bounded forcing axioms:<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc}) \iff MA_{\omega_1}$<br />
** So $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc})$ is consistent with ZFC, because Martin's axiom is consistent with ZFC.<br />
** More generally: For any ccc poset $\mathbb{P}$, $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P}) \iff MA_{\omega_1}&#40;\mathbb{P})$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper}) \iff BPFA$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper}) \iff BSPFA$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Stat-pres}) \iff BMM$<br />
<br />
Other:<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_1)$ holds &#40;as most cases with $Φ ⊆ Σ_1&#40;H&#40;ω_1))$ do, see section "Basic properties").<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ implies $\neg CH$ for any $\mathbb{P}$ that adds a real number.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba})$ is equivalent to the $ω_1$-additivity of the Lebesgue measure.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba-category})$ is equivalent to the $ω_1$-additivity of the Baire property.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ is inconsistent with ZFC for any $\mathbb{P}$ that collapses $ω_1$.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, ω_1\textit{-pres})$ is inconsistent with ZFC.<br />
* ......<br />
<br />
=== Results for $H&#40;ω_2)$ and $Σ_2$ ===<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.<br />
** Because:<br />
*** by adding $ω_1$ Cohen reals &#40;a $σ$-centered forcing notion) one adds a Luzin set &#40;an uncountable set of reals that intersects every meager set in at most a countable set; its existence is a $Σ_2$ statement in $H&#40;ω_2)$)<br />
*** and then we may iterate in length the continuum $\textit{Amoeba-category}$ &#40;another $σ$-centered forcing notion), so that in the generic extension every set of size $ω_1$ is meager.<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{Knaster})$ is false.<br />
** The above argument applies because any iteration of $\textit{Amoeba-category}$ with finite support is Knaster.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_2}, \textit{$\sigma$-centered})$ is false.<br />
** The argument applies because given any set of reals in $H&#40;ω_2)$ we can force with $\textit{Amoeba-category}$ to make it meager.<br />
<br />
=== Results for $H&#40;\kappa)$, $\kappa \ge \omega_3$ ===<br />
* ......<br />
<br />
=== Results for $L&#40;H&#40;ω_1))$ &#40;$=L&#40;\mathbb{R})$) ===<br />
&#40;$L&#40;H&#40;ω_1))=L&#40;\mathbb{R})$, because every element of $H&#40;ω_1)$ can be easily coded by a real number.)<br />
<br />
Results:<br />
* The consistency strength of $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals:<br />
** If there is a proper class of Woodin cardinals, then $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ holds.<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ implies that the [[axiom of determinacy]] holds in $L&#40;\mathbb{R})$ &#40;$\mathrm{AD}^{L&#40;\mathbb{R})}$, equiconsistent with $\mathrm{AD}$, equiconsistent with the existence of infinitely many Woodin cardinals).<br />
* If $δ$ is a weakly compact Woodin cardinal, then $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \mathbb{P})$ holds for every proper poset $\mathbb{P} ∈ V_δ$.<br />
** Therefore, $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$ follows from the existence of a proper class of weakly compact Woodin cardinals.<br />
** The existence of just a remarkable cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$.<br />
* The following are equiconsistent with the existence of a weakly compact cardinal:<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Knaster})$<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{ccc})$<br />
* The following are equiconsistent with the existence of a Mahlo cardinal:<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-centered})$<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-linked})$<br />
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.<br />
* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.<br />
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.<br />
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.<br />
<br />
=== Results for $L&#40;H&#40;ω_2))$ ===<br />
* ......<br />
<br />
== Open problems ==<br />
* Does $\mathcal{A}&#40;H&#40;ω_1), Σ_ω , Γ)$, for $Γ$ the class of Borel ccc forcing notions, imply that every [[projective]] set of real numbers is Lebesgue measurable?<br />
* ......<br />
<br />
{{References}}<br />
<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Forcing]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_cardinals&diff=4158Reflecting cardinals2022-05-14T19:56:09Z<p>BartekChom: : ''Not to be confused with reflecting ordinals.''</p>
<hr />
<div>{{DISPLAYTITLE: Reflecting cardinals}}<br />
[[Category:Middle attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting ordinal]]s.''<br />
Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak [[Kripke-Platek]] set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-separation after reflection down to a $V_\alpha$ containing the range of the defined function. Several philosophers have advanced philosophical justifications of large cardinals based on ideas arising from reflection.<br />
<br />
==Reflection theorem== <br />
The Reflection theorem is one of the most important theorems in Set Theory, being the basis for several large cardinals. The Reflection theorem is in fact a "meta-theorem," a theorem about proving theorems. The Reflection theorem intuitively encapsulates the idea that we can find sets resembling the class $V$ of all sets.<br />
<br />
'''Theorem (Reflection):''' For every set $M$ and formula $\phi(x_0...x_n,p)$ ($p$ is a parameter) there exists some limit ordinal $\alpha$ such that $V_\alpha\supseteq M$ such that $\phi^{V_\alpha}(x_0...x_n,p)\leftrightarrow \phi(x_0...x_n,p)$ (We say $V_\alpha$ reflects $\phi$). Assuming the Axiom of Choice, we can find some countable $M_0\supseteq M$ that reflects $\phi(x_0...x_n,p)$.<br />
<br />
Note that by conjunction, for any finite family of formulas $\phi_0...\phi_n$, as $V_\alpha$ reflects $\phi_0...\phi_n$ if and only if $V_\alpha$ reflects $\phi_0\land...\land\phi_n$. Another important fact is that the truth predicate for $\Sigma_n$ formulas is $\Sigma_{n+1}$, and so we can find a (Club class of) ordinals $\alpha$ such that $(V_\alpha,\in)\prec_{{T_{\Sigma_n}}\restriction{V_\alpha}} (V,\in)$, where $T_{\Sigma_n}$ is the truth predicate for $\Sigma_n$ and so $ZFC\vdash Con(ZFC(\Sigma_n))$ for every $n$, where $ZFC(\Sigma_n)$ is $ZFC$ with Replacement and Separation restricted to $\Sigma_n$.<br />
<br />
'''Lemma:''' If $W_\alpha$ is a cumulative hierarchy, there are arbitrarily large limit ordinals $\alpha$ such that $\phi^{W_\alpha}(x_0...x_n,p)\leftrightarrow \phi^W(x_0...x_n,p)$.<br />
<br />
==Reflection and correctness==<br />
<br />
For any class $\Gamma$ of formulas, an inaccessible cardinal $\kappa$ is ''$\Gamma$-reflecting'' if and only if $H_\kappa\prec_\Gamma V$, meaning that for any $\varphi\in\Gamma$ and $a\in H_\kappa$ we have $V\models\varphi[a]\iff H_\kappa\models\varphi[a]$. For example, an inaccessible cardinal is ''$\Sigma_n$-reflecting'' if and only if $H_\kappa\prec_{\Sigma_n} V$. In the case that $\kappa$ is not necessarily inaccessible, we say that $\kappa$ is ''$\Gamma$-correct'' if and only if $H_\kappa\prec_\Gamma V$''. <br />
<br />
* A simple L&ouml;wenheim-Skolem argument shows that every uncountable cardinal $\kappa$ is $\Sigma_1$-correct.<br />
* For each natural number $n$, the $\Sigma_n$-correct cardinals form a closed unbounded proper class of cardinals, as a consequence of the [[reflection theorem]]. This class is sometimes denoted by $C^{(n)}$ and the $\Sigma_n$-correct cardinals are also sometimes referred to as the $C^{(n)}$-cardinals. <br />
* Every $\Sigma_2$-correct cardinal is a [[beth fixed point | $\beth$-fixed point]] and a limit of such $\beth$-fixed points, as well as an [[aleph | $\aleph$-fixed point]] and a limit of such. Consequently, we may equivalently define for $n\geq 2$ that $\kappa$ is $\Sigma_n$-correct if and only if $V_\kappa\prec_{\Sigma_n} V$. <br />
<br />
A cardinal $\kappa$ is ''correct'', written $V_\kappa\prec V$, if it is $\Sigma_n$-correct for each $n$. This is not expressible by a single assertion in the language of set theory (since if it were, the least such $\kappa$ would have to have a smaller one inside $V_\kappa$ by elementarity). Nevertheless, $V_\kappa\prec V$ is expressible as a scheme in the language of set theory with a parameter (or constant symbol) for $\kappa$. <br />
<br />
Although it may be surprising, the existence of a correct cardinal is equiconsistent with ZFC. This can be seen by a simple compactness argument, using the fact that the theory ZFC+"$\kappa$ is correct" is finitely consistent, if ZFC is consistent, precisely by the observation about $\Sigma_n$-correct cardinals above.<br />
<br />
[[C^(n)|$C^{(n)}$]] are the classes of $\Sigma_n$-correct ordinals. These classes are clubs (closed unbounded). $C^{(0)}$ is the class of all ordinals. $C^{(1)}$ is precisely the class of all uncountable cardinals $α$ such that $V_\alpha=H(\alpha)$; i.e. precisely the Beth fixed points. References to the $C^{(n)}$ classes (different from just the requirement that the cardinal belongs to $C^{(n)}$) can sometimes make large cardinal properties stronger (for example $C^{(n)}$-[[superstrong]], $C^{(n)}$-[[supercompact]], $C^{(n)}$-[[extendible]], $C^{(n)}$-[[huge]] and $C^{(n)}$-[[rank-into-rank]] cardinals). On the other hand, every [[measurable]] cardinal is $C^{(n)}$-measurable for all $n$ and every ($λ$-)[[strong]] cardinal is ($λ$-)$C^{(n)}$-strong for all $n$.<cite>Bagaria2012:CnCardinals</cite><br />
<br />
A cardinal $\kappa$ is ''reflecting'' if it is inaccessible and correct. Just as with the notion of correctness, this is not first-order expressible as a single assertion in the language of set theory, but it is expressible as a scheme (''Lévy scheme''). The existence of such a cardinal is equiconsistent to the assertion [[ORD is Mahlo]].<br />
<br />
If there is a pseudo [[uplifting]] cardinal, or indeed, merely a pseudo $0$-uplifting cardinal $\kappa$, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. You can get this by taking some $\lambda\gt\kappa$ such that $V_\kappa\prec V_\lambda$.<br />
<br />
== $\Sigma_2$-correct cardinals == <br />
<br />
The $\Sigma_2$-correct cardinals are a particularly useful and robust class of cardinals, because of the following characterization: $\kappa$ is $\Sigma_2$-correct if and only if for any $x\in V_\kappa$ and any formula $\varphi$ of any complexity, whenever there is an ordinal $\alpha$ such that $V_\alpha\models\varphi[x]$, then there is $\alpha\lt\kappa$ with $V_\alpha\models\varphi[x]$. The reason this is equivalent to $\Sigma_2$-correctness is that assertions of the form $\exists \alpha\ V_\alpha\models\varphi(x)$ have complexity $\Sigma_2(x)$, and conversely all $\Sigma_2(x)$ assertions can be made in that form. <br />
<br />
It follows, for example, that if $\kappa$ is $\Sigma_2$-correct, then any feature of $\kappa$ or any larger cardinal than $\kappa$ that can be verified in a large $V_\alpha$ will reflect below $\kappa$. So if $\kappa$ is $\Sigma_2$-reflecting, for example, then there must be unboundedly many inaccessible cardinals below $\kappa$. Similarly, if $\kappa$ is $\Sigma_2$-reflecting and measurable, then there must be unboundedly many measurable cardinals below $\kappa$.<br />
<br />
One can also say that a $Σ_2$-reflecting cardinal is a regular cardinal $κ$ such that $V_κ \preccurlyeq_{Σ_2} V$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
Other facts:<br />
* [[Remarkable]] cardinals are $Σ_2$-reflecting.<cite>Wilson2018:WeaklyRemarkableCardinals</cite><br />
* It is relatively consistent that ZFC and the [[Vopenka|generic Vopěnka scheme]] holds, yet [[Ord is Mahlo|$Ord$ is not definably Mahlo]] and not even $∆_2$-Mahlo. In such a model, there can be no $Σ_2$-reflecting cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* An [[Axioms of generic absoluteness|axiom of generic absoluteness]], $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$, is equiconsistent with the existence of a $Σ_2$-reflecting cardinal.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
As for $\Sigma_3$-correctness, $\Sigma_3$-correct cardinals (among others) cannot be Laver indestructible, because $\Sigma_3$-[[extendible]] cardinals cannot.<cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite><br />
<br />
==The Feferman theory==<br />
<br />
This is the theory, expressed in the language of set theory augmented with a new unary class predicate symbol $C$, asserting that $C$ is a closed unbounded class of cardinals, and every $\gamma\in C$ has $V_\gamma\prec V$. In other words, the theory consists of the following scheme of assertions: $$\forall\gamma\in C\ \forall x\in V_\gamma\ \bigl[\varphi(x)\iff\varphi^{V_\gamma}(x)\bigr]$$<br />
as $\varphi$ ranges over all formulas. Thus, the Feferman theory asserts that the universe $V$ is the union of a chain of elementary substructures $$V_{\gamma_0}\prec V_{\gamma_1}\prec\cdots\prec V_{\gamma_\alpha}\prec\cdots \prec V$$<br />
Although this may appear at first to be a rather strong theory, since it seems to imply at the very least that each $V_\gamma$ for $\gamma\in C$ is a model of ZFC, this conclusion would be incorrect. In fact, the theory does ''not'' imply that any $V_\gamma$ is a model of ZFC, and does not prove $\text{Con}(\text{ZFC})$; rather, the theory implies for each axiom of ZFC separately that each $V_\gamma$ for $\gamma\in C$ satisfies it. Since the theory is a scheme, there is no way to prove from that theory that any particular $\gamma\in C$ has $V_\gamma$ satisfying more than finitely many axioms of ZFC. In particular, a simple compactness argument shows that the Feferman theory is consistent provided only that ZFC itself is consistent, since any finite subtheory of the Feferman theory is true by the [[reflection theorem]] in any model of ZFC. It follows that the Feferman theory is actually conservative over ZFC, and proves with ZFC no new facts about sets that is not already provable in ZFC alone. <br />
<br />
The Feferman theory was proposed as a natural theory in which to undertake the category-theoretic uses of [[Grothendieck universe | Grothendieck universes]], but without the large cardinal penalty of a proper class of inaccessible cardinals. Indeed, the Feferman theory offers the advantage that the universes are each elementary substructures of one another, which is a feature not generally true under the [[universe axiom]].<br />
<br />
==Maximality Principle==<br />
<br />
The existence of an inaccessible reflecting cardinal is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if $\kappa$ is an inaccessible reflecting cardinal, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model with an inaccessible reflecting cardinal.<br />
<br />
== $Σ_n(A)$-correct ==<br />
(this section from <cite>Hamkins2016:TheVopenkaPrincipleIs</cite>)<br />
<br />
Definitions:<br />
* An ordinal $γ$ is $Σ_n(A)$-correct, if $⟨V_γ, ∈, A ∩ V_γ⟩ ≺_{Σ_n} ⟨V, ∈, A⟩$.<br />
* A cardinal $κ$ is $Σ_n(A)$-reflecting, if it is inaccessible and $Σ_n(A)$-correct.<br />
<br />
Results:<br />
* If $κ$ is $A$-[[extendible]] for a class $A$, then $κ$ is $Σ_2(A)$-reflecting.<br />
* If $κ$ is $A ⊕ C$-extendible, where $C$ is the class of all $Σ_1(A)$-correct ordinals, then $κ$ is $Σ_3(A)$-reflecting.<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Axioms_of_generic_absoluteness&diff=4157Axioms of generic absoluteness2022-05-14T19:51:01Z<p>BartekChom: reflecting</p>
<hr />
<div>&#40;from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>; compare [[Projective#Generically_absolute]])<br />
<br />
'''Axioms of generic absoluteness''' are axioms $\mathcal{A}&#40;W, \Phi, \Gamma)$ of the form “$W$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ \Gamma$”, where<br />
* $W$ is a subclass of $V$.<br />
* $\Phi$ is a class of sentences.<br />
* $\Gamma$ is a class of [[forcing]] notions.<br />
* “$W^V$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$” &#40;symbolically $W^V \equiv_\Phi W^{V^\mathbb{P}}$) means that $\forall_{\phi\in\Phi} &#40;W^V \models \phi \quad \text{iff} \quad W^{V^\mathbb{P}} \models \phi)$.<br />
$W$$, \Phi$ and $\Gamma$ must be definable classes for $\mathcal{A}&#40;W, \Phi, \Gamma)$ to be a sentence in the first-order language of Set Theory.<br />
<br />
== Notation ==<br />
* If $Γ$ contains only one element, $\mathbb{P}$, then one can write $\mathcal{A}&#40;W, Φ, \mathbb{P})$ instead of $\mathcal{A}&#40;W, Φ, Γ)$.<br />
* If $Γ$ is the class of all set-forcing notions, then one can just write $\mathcal{A}&#40;W, Φ)$.<br />
* The class of $\Sigma_n$ sentences with parameters from $W$ is denoted $\Sigma_n&#40;W)$ or in short $\underset{\sim}{\Sigma_n}$.<br />
** Analogously for $\Pi_n$ etc.<br />
** Boldface $\mathbf{\Sigma_n}$ is used in other sources for similar notions.<br />
<br />
== Basic properties ==<br />
* If $Φ ⊆ Φ_0$ and $Γ ⊆ Γ_0$, then $\mathcal{A}&#40;W, Φ_0 , Γ_0)$ implies $\mathcal{A}&#40;W, Φ, Γ)$.<br />
* $\mathcal{A}&#40;W, Φ, Γ)$ is equivalent to $\mathcal{A}&#40;W, \bar{Φ}, Γ)$, where $\bar{Φ}$ is the closure of $Φ$ under finite Boolean combinations.<br />
** Eg. $\mathcal{A}&#40;W, Σ_n , Γ)$ is equivalent to $\mathcal{A}&#40;W, Π_n , Γ)$<br />
* If $Φ ⊆ \underset{\sim}{Σ_0}$, then $\mathcal{A}&#40;W, Φ, Γ)$ holds for all transitive $W$ and all $Γ$ such that $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.<br />
* If $Φ ⊆ Σ_1&#40;H&#40;ω_1))$, then &#40;by the Levy-Shoenfield absoluteness theorem) $\mathcal{A}&#40;W, Φ, Γ)$ holds for every transitive model $W$ of a weak fragment of ZF that contains the parameters of $Φ$, and all $Γ$, provided $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.<br />
** In particular, the following hold:<br />
*** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_1})$<br />
*** $\mathcal{A}&#40;H&#40;κ), Σ_1&#40;H&#40;ω_1)))$ for $κ > ω_1$<br />
*** $\mathcal{A}&#40;V, Σ_1&#40;H&#40;ω_1)))$<br />
* $\mathcal{A}&#40;V, \underset{\sim}{Σ_1}, \mathbb{P})$ fails for any nontrivial $\mathbb{P}$ &#40;any $\mathbb{P}$ that adds some new set), because a proper class $W \neq V$ can never be an elementary substructure of $V$, since otherwise, by elementarity, $V_α^W = V_α$ for every ordinal $α$ and so $W = V$ &#40;contradiction).<br />
* For every forcing notion $\mathbb{P}$, $L^V = L^{V^\mathbb{P}}$, so $A&#40;L, Φ)$ holds for all $Φ$.<br />
* For every forcing notion $\mathbb{P}$, $H&#40;ω)^V = H&#40;ω)^{V^\mathbb{P}}$, so $A&#40;H&#40;ω), Φ)$ holds for all $Φ$.<br />
<br />
We see that, when $W = V$, $\Phi$ is the class of all sentences or $\Gamma$ is the class of all forcing notions, then the other two must be very small for the axiom to be consistent with ZFC.<br />
<br />
== Results ==<br />
Interesting results are obtained for $W = H&#40;κ)$ or $W = L&#40;H&#40;κ))$ with some definable uncountable cardinal $κ$.<br />
* $H&#40;κ)$ is better then $V_\alpha$ &#40;for ordinal $\alpha$), because<br />
** for regular $κ$ it is a model of ZFC without powerset and so it satisfies replacement.<br />
*** This allows for nice properties like: if $\mathbb{P} ∈ H&#40;κ)$, then a filter $G ⊆ \mathbb{P}$ is generic over $V$ iff it is generic over $H&#40;κ)$.<br />
** If $κ < λ$, then $\mathcal{A}&#40;H&#40;λ), \underset{\sim}{Σ_1}, Γ)$ implies $\mathcal{A}&#40;H&#40;κ), \underset{\sim}{Σ_1}, Γ)$.<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_2$ ===<br />
Relations with large cardinal properties:<br />
* If [[zero sharp|$X^\sharp$]] &#40;[[zero sharp|sharp]]) exists for every set $X$, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ holds.<br />
* The following are equiconsistent with ZFC:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_2)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ does not imply that $ω_1^L$ is countable.<br />
* If $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ holds after forcing with a certain proper poset, then either $ω_1$ is [[Mahlo]] in $L$ or $ω_2$ is [[inaccessible]] in $L$.<br />
* The following are equiconsistent with the existence of a $Σ_2$-[[reflecting cardinals|reflecting]] cardinal:<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$<br />
*** &#40;Because $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$ implies that $ω_1$ is a $Σ_2$-reflecting cardinal in $L[x]$ for every real $x$.)<br />
<br />
Relations with bounded forcing axioms:<br />
* $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$.<br />
* $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$.<br />
* $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.<br />
* $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.<br />
* The last four implications cannot be reversed, because all axioms of the form $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Stat-pres})$ are preserved after collapsing the continuum to $ω_1$ by $σ$-closed forcing and so are all consistent with CH and do not imply any of the bounded forcing axioms.<br />
* If $θ$ is the statement that every subset of $ω_1$ is constructible from a real, that is, for every $X ⊆ ω_1$ there is $x ⊆ ω$ with $X ∈ L[x]$ and<br />
** $ω_1$ is not [[weakly compact]] in $L[x]$ for some $x ⊆ ω$, then:<br />
** $MA_{ω_1}$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$ plus $θ$.<br />
** $BPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ plus $θ$.<br />
** $BSPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ plus $θ$.<br />
* $BSPFA$ is consistent with $ω_1^L = ω_1$.<br />
* $BMM$ implies that $ω_1$ is weakly compact in $L[x]$ for every $x ⊆ ω$.<br />
<br />
Equivalences to other statements:<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ and $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba-category})$ are both equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals has the property of Baire.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Lebesgue measurable.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Mathias})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Ramsey.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ is equivalent to the statement that every $\underset{\sim}{\Delta^1_2}$ set of reals is [[Projective#Suslin_sets_and_universally_Baire_sets|universally Baire]].<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_3$ ===<br />
Relations with large cardinal properties:<br />
* Each of the following implies that $ω_1$ is inaccessible in $L[x]$ for every real $x$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Cohen})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Hechler})$<br />
* The following are equiconsistent with the existence of a sharp for each set:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Stat-pres})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{$ω_1$-pres})$ &#40;obviously from the other two)<br />
* The existence of a $Σ_2$-reflecting cardinal and a sharp for each set is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$.<br />
* The following are equiconsistent with the existence of a weakly compact cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Knaster})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{ccc})$<br />
* The following are equiconsistent with the existence of a Mahlo cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-centered})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-linked})$<br />
* The following are equiconsistent with the existence of an inaccessible cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of posets that are absolutely-ccc and strongly-$\underset{\sim}{Σ_2}$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of strongly-proper posets that are $Σ_2$ definable in $H&#40;ω_1)$ with parameters<br />
** &#40;This result is optimal, for there is a, provably in ZFC, ccc poset $\mathbb{P}$ which is both $Σ_2$ and $Π_2$ definable in $H&#40;ω_1)$, without parameters, and for which the axiom $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \mathbb{P})$ fails if $ω_1$ is not a $Π_1$-Mahlo cardinal in $L$.)<br />
* The following are equiconsistent with the existence of a [[remarkable]] cardinal for $3 \le n \le \omega$:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Proper})$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Proper})$<br />
<br />
Relations with bounded forcing axioms:<br />
* If $x^\sharp$ exists for every real $x$ and the second uniform indiscernible is $< ω_2$, then<br />
** $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{ccc})$.<br />
** $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Proper})$.<br />
** $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Semi-proper})$.<br />
** $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Stat-pres})$.<br />
<br />
Relations with other statements:<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba-category}, \textit{Cohen}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals has the property of Baire.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba}, \textit{Random}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is Lebesgue measurable.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$ implies that every $\underset{\sim}{\Sigma^1_3}$ set of reals is Ramsey.<br />
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is universally Baire.<br />
** The converse does not hold.<br />
<br />
=== Results for $H&#40;ω_1)$ and $Σ_n$, $4 \le n \le ω$ ===<br />
* $\mathcal{A}&#40;H&#40;ω_1), Σ_4)$ implies that [[zero dagger|$X^\dagger$ &#40;dagger)]] exists for every set $X$.<br />
* The following are equiconsistent with the existence of infinitely many [[strong]] cardinals:<br />
** $\mathcal{A}&#40;H&#40;ω_1), Σ_ω)$<br />
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$<br />
* If there is a proper class of [[Woodin]] cardinals, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$.<br />
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.<br />
* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.<br />
* The consistency strength of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n})$ for $n \ge 4$ is<br />
** at least that of $n-3$ strong cardinals<br />
** and at most that of $n-3$ strong cardinals with a $Σ_2$-reflecting cardinal above them.<br />
<br />
=== Results for $H&#40;ω_2)$ and $Σ_1$ ===<br />
Relations with large cardinal properties:<br />
* The following are equiconsistent with the existence of a $Σ_2$-reflecting cardinal:<br />
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper})$<br />
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper})$<br />
<br />
Equivalence to bounded forcing axioms:<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc}) \iff MA_{\omega_1}$<br />
** So $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc})$ is consistent with ZFC, because Martin's axiom is consistent with ZFC.<br />
** More generally: For any ccc poset $\mathbb{P}$, $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P}) \iff MA_{\omega_1}&#40;\mathbb{P})$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper}) \iff BPFA$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper}) \iff BSPFA$<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper}) \iff BMM$<br />
<br />
Other:<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_1)$ holds &#40;as most cases with $Φ ⊆ Σ_1&#40;H&#40;ω_1))$ do, see section "Basic properties").<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ implies $\neg CH$, for any $\mathbb{P}$ that adds a real number.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba})$ is equivalent to the $ω_1$-additivity of the Lebesgue measure.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba-category})$ is equivalent to the $ω_1$-additivity of the Baire property.<br />
* ......<br />
<br />
=== Results for $H&#40;ω_2)$ and $Σ_2$ ===<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.<br />
** Because:<br />
*** by adding $ω_1$ Cohen reals &#40;a $σ$-centered forcing notion) one adds a Luzin set &#40;an uncountable set of reals that intersects every meager set in at most a countable set; its existence is a $Σ_2$ statement in $H&#40;ω_2)$)<br />
*** and then we may iterate in length the continuum $\textit{Amoeba-category}$ &#40;another $σ$-centered forcing notion), so that in the generic extension every set of size $ω_1$ is meager.<br />
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{Knaster})$ is false.<br />
** The above argument applies because any iteration of $\textit{Amoeba-category}$ with finite support is Knaster.<br />
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_2}, \textit{$\sigma$-centered})$ is false.<br />
** The argument applies because given any set of reals in $H&#40;ω_2)$ we can force with $\textit{Amoeba-category}$ to make it meager.<br />
<br />
=== Results for $H&#40;\kappa)$, $\kappa \ge \omega_3$ ===<br />
* ......<br />
<br />
=== Results for $L&#40;H&#40;ω_1))$ &#40;$=L&#40;\mathbb{R})$) ===<br />
&#40;$L&#40;H&#40;ω_1))=L&#40;\mathbb{R})$, because every element of $H&#40;ω_1)$ can be easily coded by a real number.)<br />
<br />
Results:<br />
* The consistency strength of $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals:<br />
** If there is a proper class of Woodin cardinals, then $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ holds.<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ implies that the [[axiom of determinacy]] holds in $L&#40;\mathbb{R})$ &#40;$\mathrm{AD}^{L&#40;\mathbb{R})}$, equiconsistent with $\mathrm{AD}$, equiconsistent with the existence of infinitely many Woodin cardinals).<br />
* If $δ$ is a weakly compact Woodin cardinal, then $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \mathbb{P})$ holds for every proper poset $\mathbb{P} ∈ V_δ$.<br />
** Therefore, $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$ follows from the existence of a proper class of weakly compact Woodin cardinals.<br />
** The existence of just a remarkable cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$.<br />
* The following are equiconsistent with the existence of a weakly compact cardinal:<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Knaster})$<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{ccc})$<br />
* The following are equiconsistent with the existence of a Mahlo cardinal:<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-centered})$<br />
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-linked})$<br />
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.<br />
* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.<br />
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.<br />
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.<br />
<br />
=== Results for $L&#40;H&#40;ω_2))$ ===<br />
* ......<br />
<br />
== Open problems ==<br />
* Does $\mathcal{A}&#40;H&#40;ω_1), Σ_ω , Γ)$, for $Γ$ the class of Borel ccc forcing notions, imply that every [[projective]] set of real numbers is Lebesgue measurable?<br />
* ......<br />
<br />
{{References}}<br />
<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Forcing]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_cardinals&diff=4156Reflecting cardinals2022-05-14T19:47:44Z<p>BartekChom: /* $\Sigma_2$-correct cardinals */ -[[]]</p>
<hr />
<div>{{DISPLAYTITLE: Reflecting cardinals}}<br />
[[Category:Middle attic]]<br />
[[Category:Reflection principles]]<br />
Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak [[Kripke-Platek]] set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-separation after reflection down to a $V_\alpha$ containing the range of the defined function. Several philosophers have advanced philosophical justifications of large cardinals based on ideas arising from reflection. <br />
<br />
==Reflection theorem== <br />
The Reflection theorem is one of the most important theorems in Set Theory, being the basis for several large cardinals. The Reflection theorem is in fact a "meta-theorem," a theorem about proving theorems. The Reflection theorem intuitively encapsulates the idea that we can find sets resembling the class $V$ of all sets.<br />
<br />
'''Theorem (Reflection):''' For every set $M$ and formula $\phi(x_0...x_n,p)$ ($p$ is a parameter) there exists some limit ordinal $\alpha$ such that $V_\alpha\supseteq M$ such that $\phi^{V_\alpha}(x_0...x_n,p)\leftrightarrow \phi(x_0...x_n,p)$ (We say $V_\alpha$ reflects $\phi$). Assuming the Axiom of Choice, we can find some countable $M_0\supseteq M$ that reflects $\phi(x_0...x_n,p)$.<br />
<br />
Note that by conjunction, for any finite family of formulas $\phi_0...\phi_n$, as $V_\alpha$ reflects $\phi_0...\phi_n$ if and only if $V_\alpha$ reflects $\phi_0\land...\land\phi_n$. Another important fact is that the truth predicate for $\Sigma_n$ formulas is $\Sigma_{n+1}$, and so we can find a (Club class of) ordinals $\alpha$ such that $(V_\alpha,\in)\prec_{{T_{\Sigma_n}}\restriction{V_\alpha}} (V,\in)$, where $T_{\Sigma_n}$ is the truth predicate for $\Sigma_n$ and so $ZFC\vdash Con(ZFC(\Sigma_n))$ for every $n$, where $ZFC(\Sigma_n)$ is $ZFC$ with Replacement and Separation restricted to $\Sigma_n$.<br />
<br />
'''Lemma:''' If $W_\alpha$ is a cumulative hierarchy, there are arbitrarily large limit ordinals $\alpha$ such that $\phi^{W_\alpha}(x_0...x_n,p)\leftrightarrow \phi^W(x_0...x_n,p)$.<br />
<br />
==Reflection and correctness==<br />
<br />
For any class $\Gamma$ of formulas, an inaccessible cardinal $\kappa$ is ''$\Gamma$-reflecting'' if and only if $H_\kappa\prec_\Gamma V$, meaning that for any $\varphi\in\Gamma$ and $a\in H_\kappa$ we have $V\models\varphi[a]\iff H_\kappa\models\varphi[a]$. For example, an inaccessible cardinal is ''$\Sigma_n$-reflecting'' if and only if $H_\kappa\prec_{\Sigma_n} V$. In the case that $\kappa$ is not necessarily inaccessible, we say that $\kappa$ is ''$\Gamma$-correct'' if and only if $H_\kappa\prec_\Gamma V$''. <br />
<br />
* A simple L&ouml;wenheim-Skolem argument shows that every uncountable cardinal $\kappa$ is $\Sigma_1$-correct.<br />
* For each natural number $n$, the $\Sigma_n$-correct cardinals form a closed unbounded proper class of cardinals, as a consequence of the [[reflection theorem]]. This class is sometimes denoted by $C^{(n)}$ and the $\Sigma_n$-correct cardinals are also sometimes referred to as the $C^{(n)}$-cardinals. <br />
* Every $\Sigma_2$-correct cardinal is a [[beth fixed point | $\beth$-fixed point]] and a limit of such $\beth$-fixed points, as well as an [[aleph | $\aleph$-fixed point]] and a limit of such. Consequently, we may equivalently define for $n\geq 2$ that $\kappa$ is $\Sigma_n$-correct if and only if $V_\kappa\prec_{\Sigma_n} V$. <br />
<br />
A cardinal $\kappa$ is ''correct'', written $V_\kappa\prec V$, if it is $\Sigma_n$-correct for each $n$. This is not expressible by a single assertion in the language of set theory (since if it were, the least such $\kappa$ would have to have a smaller one inside $V_\kappa$ by elementarity). Nevertheless, $V_\kappa\prec V$ is expressible as a scheme in the language of set theory with a parameter (or constant symbol) for $\kappa$. <br />
<br />
Although it may be surprising, the existence of a correct cardinal is equiconsistent with ZFC. This can be seen by a simple compactness argument, using the fact that the theory ZFC+"$\kappa$ is correct" is finitely consistent, if ZFC is consistent, precisely by the observation about $\Sigma_n$-correct cardinals above.<br />
<br />
[[C^(n)|$C^{(n)}$]] are the classes of $\Sigma_n$-correct ordinals. These classes are clubs (closed unbounded). $C^{(0)}$ is the class of all ordinals. $C^{(1)}$ is precisely the class of all uncountable cardinals $α$ such that $V_\alpha=H(\alpha)$; i.e. precisely the Beth fixed points. References to the $C^{(n)}$ classes (different from just the requirement that the cardinal belongs to $C^{(n)}$) can sometimes make large cardinal properties stronger (for example $C^{(n)}$-[[superstrong]], $C^{(n)}$-[[supercompact]], $C^{(n)}$-[[extendible]], $C^{(n)}$-[[huge]] and $C^{(n)}$-[[rank-into-rank]] cardinals). On the other hand, every [[measurable]] cardinal is $C^{(n)}$-measurable for all $n$ and every ($λ$-)[[strong]] cardinal is ($λ$-)$C^{(n)}$-strong for all $n$.<cite>Bagaria2012:CnCardinals</cite><br />
<br />
A cardinal $\kappa$ is ''reflecting'' if it is inaccessible and correct. Just as with the notion of correctness, this is not first-order expressible as a single assertion in the language of set theory, but it is expressible as a scheme (''Lévy scheme''). The existence of such a cardinal is equiconsistent to the assertion [[ORD is Mahlo]].<br />
<br />
If there is a pseudo [[uplifting]] cardinal, or indeed, merely a pseudo $0$-uplifting cardinal $\kappa$, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. You can get this by taking some $\lambda\gt\kappa$ such that $V_\kappa\prec V_\lambda$.<br />
<br />
== $\Sigma_2$-correct cardinals == <br />
<br />
The $\Sigma_2$-correct cardinals are a particularly useful and robust class of cardinals, because of the following characterization: $\kappa$ is $\Sigma_2$-correct if and only if for any $x\in V_\kappa$ and any formula $\varphi$ of any complexity, whenever there is an ordinal $\alpha$ such that $V_\alpha\models\varphi[x]$, then there is $\alpha\lt\kappa$ with $V_\alpha\models\varphi[x]$. The reason this is equivalent to $\Sigma_2$-correctness is that assertions of the form $\exists \alpha\ V_\alpha\models\varphi(x)$ have complexity $\Sigma_2(x)$, and conversely all $\Sigma_2(x)$ assertions can be made in that form. <br />
<br />
It follows, for example, that if $\kappa$ is $\Sigma_2$-correct, then any feature of $\kappa$ or any larger cardinal than $\kappa$ that can be verified in a large $V_\alpha$ will reflect below $\kappa$. So if $\kappa$ is $\Sigma_2$-reflecting, for example, then there must be unboundedly many inaccessible cardinals below $\kappa$. Similarly, if $\kappa$ is $\Sigma_2$-reflecting and measurable, then there must be unboundedly many measurable cardinals below $\kappa$.<br />
<br />
One can also say that a $Σ_2$-reflecting cardinal is a regular cardinal $κ$ such that $V_κ \preccurlyeq_{Σ_2} V$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
Other facts:<br />
* [[Remarkable]] cardinals are $Σ_2$-reflecting.<cite>Wilson2018:WeaklyRemarkableCardinals</cite><br />
* It is relatively consistent that ZFC and the [[Vopenka|generic Vopěnka scheme]] holds, yet [[Ord is Mahlo|$Ord$ is not definably Mahlo]] and not even $∆_2$-Mahlo. In such a model, there can be no $Σ_2$-reflecting cardinals.<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* An [[Axioms of generic absoluteness|axiom of generic absoluteness]], $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$, is equiconsistent with the existence of a $Σ_2$-reflecting cardinal.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
As for $\Sigma_3$-correctness, $\Sigma_3$-correct cardinals (among others) cannot be Laver indestructible, because $\Sigma_3$-[[extendible]] cardinals cannot.<cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite><br />
<br />
==The Feferman theory==<br />
<br />
This is the theory, expressed in the language of set theory augmented with a new unary class predicate symbol $C$, asserting that $C$ is a closed unbounded class of cardinals, and every $\gamma\in C$ has $V_\gamma\prec V$. In other words, the theory consists of the following scheme of assertions: $$\forall\gamma\in C\ \forall x\in V_\gamma\ \bigl[\varphi(x)\iff\varphi^{V_\gamma}(x)\bigr]$$<br />
as $\varphi$ ranges over all formulas. Thus, the Feferman theory asserts that the universe $V$ is the union of a chain of elementary substructures $$V_{\gamma_0}\prec V_{\gamma_1}\prec\cdots\prec V_{\gamma_\alpha}\prec\cdots \prec V$$<br />
Although this may appear at first to be a rather strong theory, since it seems to imply at the very least that each $V_\gamma$ for $\gamma\in C$ is a model of ZFC, this conclusion would be incorrect. In fact, the theory does ''not'' imply that any $V_\gamma$ is a model of ZFC, and does not prove $\text{Con}(\text{ZFC})$; rather, the theory implies for each axiom of ZFC separately that each $V_\gamma$ for $\gamma\in C$ satisfies it. Since the theory is a scheme, there is no way to prove from that theory that any particular $\gamma\in C$ has $V_\gamma$ satisfying more than finitely many axioms of ZFC. In particular, a simple compactness argument shows that the Feferman theory is consistent provided only that ZFC itself is consistent, since any finite subtheory of the Feferman theory is true by the [[reflection theorem]] in any model of ZFC. It follows that the Feferman theory is actually conservative over ZFC, and proves with ZFC no new facts about sets that is not already provable in ZFC alone. <br />
<br />
The Feferman theory was proposed as a natural theory in which to undertake the category-theoretic uses of [[Grothendieck universe | Grothendieck universes]], but without the large cardinal penalty of a proper class of inaccessible cardinals. Indeed, the Feferman theory offers the advantage that the universes are each elementary substructures of one another, which is a feature not generally true under the [[universe axiom]].<br />
<br />
==Maximality Principle==<br />
<br />
The existence of an inaccessible reflecting cardinal is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if $\kappa$ is an inaccessible reflecting cardinal, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model with an inaccessible reflecting cardinal.<br />
<br />
== $Σ_n(A)$-correct ==<br />
(this section from <cite>Hamkins2016:TheVopenkaPrincipleIs</cite>)<br />
<br />
Definitions:<br />
* An ordinal $γ$ is $Σ_n(A)$-correct, if $⟨V_γ, ∈, A ∩ V_γ⟩ ≺_{Σ_n} ⟨V, ∈, A⟩$.<br />
* A cardinal $κ$ is $Σ_n(A)$-reflecting, if it is inaccessible and $Σ_n(A)$-correct.<br />
<br />
Results:<br />
* If $κ$ is $A$-[[extendible]] for a class $A$, then $κ$ is $Σ_2(A)$-reflecting.<br />
* If $κ$ is $A ⊕ C$-extendible, where $C$ is the class of all $Σ_1(A)$-correct ordinals, then $κ$ is $Σ_3(A)$-reflecting.<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=ORD_is_Mahlo&diff=4155ORD is Mahlo2022-05-14T19:43:30Z<p>BartekChom: reflecting</p>
<hr />
<div>{{DISPLAYTITLE: $\text{Ord}$ is Mahlo}}<br />
The assertion ''$\text{Ord}$ is Mahlo'' is the scheme expressing that the proper class REG consisting of all [[regular]] cardinals is a [[stationary]] proper class, meaning that it has elements from every definable &#40;with parameters) [[closed unbounded]] proper class of ordinals. In other words, the scheme asserts for every formula $\varphi$, that if for some parameter $z$ the class $\{\alpha\mid \varphi&#40;\alpha,z)\}$ is a closed unbounded class of ordinals, then it contains a regular cardinal.<br />
<br />
* If $\kappa$ is [[Mahlo]], then $V_\kappa\models\text{Ord is Mahlo}$. <br />
* Consequently, the existence of a Mahlo cardinal implies the consistency of $\text{Ord}$ is Mahlo, and the two notions are not equivalent.<br />
* Moreoever, since the ORD is Mahlo scheme is expressible as a first-order theory, it follows that whenever $V_\gamma\prec V_\kappa$, then also $V_\gamma$ satisfies the Levy scheme. <br />
* Consequently, if there is a Mahlo cardinal, then there is a club of cardinals $\gamma\lt\kappa$ for which $V_\gamma\models\text{Ord is Mahlo}$. <br />
<br />
A simple compactness argument establishes that $\text{Ord}$ is Mahlo is equiconsistent over $\text{ZFC}$ with the existence of an [[inaccessible reflecting cardinal]]. On the one hand, if $\kappa$ is an inaccessible reflecting cardinal, then since $V_\kappa\prec V$ it follows that any class club definable in $V$ with parameters below $\kappa$ will be unbounded in $\kappa$ and hence contain $\kappa$ as an element and consequently contain an inaccessible cardinal. On the other hand, if $\text{Ord}$ is Mahlo is consistent, then every finite fragment of the theory asserting that $\kappa$ is an inaccessible reflecting cardinal &#40;which is after all asserted as a scheme) is consistent, and hence by compactness the whole theory is consistent.<br />
<br />
If there is a pseudo [[uplifting]] &#40;proof in that article) cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a [[reflecting cardinals|reflecting]] cardinal and consequently also a transitive model of ZFC plus $\text{Ord}$ is Mahlo.<cite>HamkinsJohnstone:ResurrectionAxioms</cite><br />
<br />
Relation to the [[Vopenka|Vopěnka principle]]:<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite><br />
* The [[Vopenka|Vopěnka principle]] implies that $\text{Ord}$ is Mahlo: every club class contains a regular cardinal and indeed, an [[extendible]] cardinal and more.<br />
* If the Vopěnka scheme holds, then there is a class-forcing extension $V[C]$ where<br />
** $\text{Ord}$ is not Mahlo &#40;the class $C$ itself witnesses it), thus the Vopěnka principle fails,<br />
** but the extension adds no new sets, thus the Vopěnka scheme still holds and $\text{Ord}$ remains definably Mahlo.<br />
** The forcing preserves $\text{GBC}$.<br />
* It is relatively consistent that $\text{GBC}$ and the generic Vopěnka principle holds, yet $\text{Ord}$ is not Mahlo:<br />
** If $0^♯$ &#40;[[zero sharp]]) exists in $V$, then there is a class-forcing notion $\mathbb{P}$ definable in the constructible universe $L$, such that in any $L$-generic extension $L[C]$ by this forcing, $\text{GBC}$ and the generic Vopěnka principle hold, yet $\text{Ord}$ is not Mahlo.<br />
*** Proof includes a lemma stating: For any ordinal $δ$ and any natural number &#40;of the meta-theory — this lemma is a scheme) $n$, if $D_{δ,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $θ$ such that<br />
**** $L_θ ≺_{Σ_n} L$,<br />
**** $c ∩ θ$ is $L_θ$-generic for $\mathbb{P}^{L_θ}$ and<br />
**** in some forcing extension of $L$, there is an elementary embedding<br />
****: $j : ⟨ L_θ , ∈, c ∩ θ ⟩ → ⟨ L_θ , ∈, c ∩ θ ⟩$<br />
****: with critical point above $δ$,<br />
***: then $D_{δ,n}$ is a definable dense subclass of $\mathbb{P}$ in $L$.<br />
* It is relatively consistent that $\text{ZFC}$ and the generic Vopěnka scheme holds, yet $\text{Ord}$ is not definably Mahlo and not even $∆_2$-Mahlo:<br />
** If $0^♯$ exists in $V$, then there is a definable class-forcing notion in $L$, such that in the corresponding $L$-generic extension, $\text{GBC}$ holds, the generic Vopěnka scheme holds, but $\text{Ord}$ is not definably Mahlo, because there is a $∆_2$-definable club class avoiding the regular cardinals.<br />
** In such a model, there can be no $Σ_2$-reflecting cardinals and therefore also no [[remarkable]] cardinals.<br />
<br />
{{references}}<br />
<br />
[[Category:Large_cardinal_axioms]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Upper_attic&diff=4154Upper attic2022-05-14T19:42:10Z<p>BartekChom: links</p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg|thumb|Cape Pogue Lighthouse photo by Timothy Valentine]]<br />
[[Category:Large cardinal axioms]]<br />
<br />
Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency. <br />
<br />
* [[Berkeley]] cardinal, [[Berkeley|club Berkeley]], [[Berkeley|limit club Berkeley]] cardinal<br />
* [[Reinhardt|weakly Reinhardt]], [[Reinhardt]], [[Reinhardt|super Reinhardt]], [[Reinhardt|totally Reinhardt]] cardinal<br />
* the '''[[Kunen inconsistency]]'''<br />
* '''[[rank into rank]]''' axioms &#40;$I3$=$E_0$, $IE^\omega$, $IE$, $I2$=$E_1$, $E_i$, $I1$=$E_ω$ plus $m$-$C^{&#40;n)}$-$E_i$), [[N-fold_variants#.24.5Comega.24-fold_variants|$\omega$-fold variants]], the '''[[L of V_lambda+1|I0 axiom]]''' and its strengthenings<br />
* The [[wholeness axioms]], [[I4|axioms $\mathrm{I}_4^n$]]<br />
* [[n-fold variants|$n$-fold variants]] of hugeness &#40;plus $C^{&#40;n)}$ variants), extendibility, supercompactness, strongness, etc...<br />
* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal<br />
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal<br />
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]<br />
* [[Vopenka|Vopěnka scheme]], '''[[Vopenka|Vopěnka principle]]''', [[Vopenka#Vopěnka cardinals|Vopěnka-scheme]] cardinal, [[Vopenka#Vopěnka cardinals|Vopěnka]] &#40;=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal<br />
* [[extendible|$\alpha$-extendible]], '''[[extendible]]''', [[extendible|$C^{&#40;n)}$-extendible]], [[extendible|$A$-extendible]] cardinals<br />
* [[Woodin|Woodin for strong compactness]]<br />
<!--* [[grand reflection]] cardinal--><br />
* [[Supercompact#Enhanced supercompact cardinals|enhanced $\lambda$-supercompact]] cardinals, [[Supercompact#Enhanced supercompact cardinals|enhanced supercompact]] cardinal, [[hypercompact|$\lambda$-hypercompact]] cardinals, [[hypercompact]] cardinal<br />
* [[supercompact|$\lambda$-supercompact]] cardinals, '''[[supercompact]]''' cardinal, [[supercompact|$C^{&#40;n)}$-supercompact]] cardinals<br />
* [[strongly compact|$\lambda$-strongly compact]] cardinals, '''[[strongly compact]]''' cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact|nearly strongly compact]] cardinals<br />
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal<br />
* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal, [[superstrong|$C^{&#40;n)}$-superstrong]] hierarchy<br />
* [[Woodin|weakly hyper-Woodin]] cardinal, [[Shelah]] cardinal, [[Woodin|hyper-Woodin]] cardinal<br />
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br />
* '''[[Woodin]]''' cardinal<br />
* [[strongly tall]] cardinal<br />
* the [[strong|$\theta$-strong]], [[strong#Hypermeasurable|hypermeasurability]], [[tall|$\theta$-tall]], [[strong|$\theta$-$A$-strong]], [[tall]], '''[[strong]]''', [[strong|$A$-strong]] cardinals<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank|$o&#40;\kappa)=1$]], [[Mitchell rank|$o&#40;\kappa)=\kappa^{++}$]] <br />
*[[zero dagger| $0^\dagger$]] &#40;''zero-dagger'')<br />
* [[weakly measurable]] cardinal, '''[[measurable]]''' cardinal<br />
** singular [[Jonsson|Jónsson]] cardinal<br />
** $κ^+$-[[filter property]], [[Ramsey|strategic $&#40;\omega+1)$-Ramsey]] cardinal, [[Ramsey|strategic fully Ramsey]] cardinal, [[Ramsey|$ω_1$-very Ramsey]] cardinal, [[Ramsey|$κ$-very Ramsey]] cardinal<br />
* $κ$-[[filter property]], [[Ramsey|fully Ramsey]] &#40;=[[Ramsey|$κ$-Ramsey]]) cardinal <br />
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal, [[Ramsey|strongly Ramsey M-rank]], [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal, [[Ramsey|super Ramsey M-rank]]<br />
* $\alpha$-[[filter property]], [[Ramsey|$\alpha$-Ramsey]] cardinal &#40;for $\omega < \alpha < \kappa$), [[Ramsey|almost fully Ramsey]] &#40;=[[Ramsey|$<κ$-Ramsey]]) cardinal <br />
* [[Ramsey|$\Pi_\alpha$-Ramsey]], [[Ramsey|completely Romsey]] &#40;=[[Ramsey|$ω$-very Ramsey]]), [[Ramsey|$\alpha$-hyper completely Romsey]], [[Ramsey|super completely Romsey]] cardinals<br />
* [[Ramsey|$\alpha$-Mahlo–Ramsey]] hierarchy<br />
* [[Ramsey|Ramsey M-rank]]<br />
* [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal<br />
* [[Erdos|$\alpha$-weakly Erdős]] cardinals, [[Erdos|greatly Erdős]] cardinal<br />
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal<br />
* [[Erdos|$\omega_1$-Erdős]] cardinal and [[Erdos|$\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]<br />
* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal, [[Ramsey|$&#40;\omega, \omega_1)$-Ramsey]] cardinal<br />
* '''[[zero sharp|$0^\sharp$]] &#40;''zero-sharp'')''', existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]<br />
* [[Silver cardinal]]<br />
* the [[Erdos|$\alpha$-'''Erdős''']], [[Ramsey#.24.5Calpha.24-iterable cardinal|$\alpha$-iterable]] and [[Ramsey|$&#40;\omega, \alpha)$-Ramsey]] hierarchy for countable infinite $\alpha$<br />
* [[Erdos|$\omega$-Erdős]] cardinal, [[remarkable|weakly remarkable]] cardinal that is not remarkable<br />
* [[rank into rank|virtually rank-into-rank]] cardinal<br />
* the [[Ramsey#.24.5Calpha.24-iterable cardinal|$n$-iterable]] and [[huge|virtually $n$-huge*]] hierarchy<br />
* [[Woodin|virtually Shelah for supercompactness]] cardinal<br />
* [[extendible|virtually extendible]] &#40;=[[remarkable|$2$-remarkable]]), [[extendible|virtually $C^{&#40;n)}$-extendible]] &#40;=[[remarkable|$n+1$-remarkable]]) cardinals, [[remarkable|completely remarkable]] cardinal, [[Vopenka|Generic Vopěnka's Principle]]<br />
* [[remarkable|&#40;$1$-)'''remarkable''']] &#40;=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals, [[proper forcing axiom|weak Proper Forcing Axiom]]<br />
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] &#40;=$1$-iterable) cardinal, [[Ramsey|super weakly Ramsey]] cardinals, [[Ramsey|$\omega$-Ramsey]] cardinal<br />
* [[completely ineffable]] cardinal &#40;= $\omega$-[[filter property]])<br />
* [[Basic Theory of Elementary Embeddings]] &#40;[[BTEE|$\mathrm{BTEE}$]])<br />
* [[ineffable#Helix|the $n$-subtle, $n$-almost ineffable, $n$-ineffable cardinals' hierarchy]]<br />
* [[Ramsey|$n$-Ramsey]], [[Ramsey|genuine $n$-Ramsey]], [[Ramsey|normal $n$-Ramsey]], [[Ramsey|$<\omega$-Ramsey]] cardinals<br />
* [[weakly ineffable]] &#40;=almost ineffable=genuine $0$-[[Ramsey]]) cardinal, [[ineffable]] &#40;=normal $0$-[[Ramsey]]) cardinal<br />
* [[subtle]] cardinal<br />
* [[ineffable#Ethereal cardinal|ethereal]] cardinal<br />
* [[uplifting#Strongly Uplifting|strongly uplifting]] &#40;=[[unfoldable#Superstrongly Unfoldable|superstrongly unfoldable]]) cardinal<br />
* [[weakly superstrong]] cardinal<br />
* [[shrewd|$η$-shrewd]], [[unfoldable]], [[shrewd]] &#40;=[[unfoldable#Strongly Unfoldable|strongly unfoldable]] cardinal), [[shrewd|$\mathcal{A}$-$η$-$\mathcal{F}$-shrewd]], [[shrewd|$\mathcal{A}$-$η$-shrewd]], [[shrewd|$\mathcal{A}$-shrewd]] cardinals<br />
* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals<br />
* uncountable cardinal with the [[tree property]], '''[[weakly compact]]''' &#40;=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal<br />
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$<br />
* [[Mahlo#Hyper-Mahlo|$1$-Mahlo]], the [[Mahlo#Hyper-Mahlo|$\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo|hyper-Mahlo]] cardinals, [[Mahlo|$Ω^α$-Mahlo]] cardinals<br />
* [[Mahlo|weakly Mahlo]] cardinal, &#40;strongly) '''[[Mahlo]]''' cardinal<br />
* [[uplifting#pseudo uplifting cardinal|pseudo uplifting]] cardinal, [[uplifting]] cardinal<br />
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<!-- apparently equiconsistent with a reflecting cardinal --><br />
* [[reflecting cardinals#.24.5CSigma_2.24-correct_cardinals|$\Sigma_2$-reflecting]], [[reflecting cardinals|$\Sigma_n$-reflecting]] and [[reflecting cardinals]]<br />
* [[Mahlo|$\Sigma_n$-Mahlo]], [[weakly compact|$\Sigma_n$-weakly compact]], [[Mahlo|$\Sigma_\omega$-Mahlo]] and [[weakly compact|$\Sigma_\omega$-weakly compact]]<!-- Really? In particular, are $\Sigma_\omega$ variants not stronger then ORD is Mahlo? Maybe $\Sigma_\omega$-weakly compact is even stronger than Mahlo? --> cardinals<br />
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]] <br />
* [[inaccessible#Degrees of inaccessibility|$1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility|$\alpha$-inaccessible]] hierarchy, [[inaccessible#Hyper-inaccessible|hyper-inaccessible]] cardinals, [[inaccessible|$Ω^α$-inaccessible]] cardinals<br />
* [[inaccessible#Universes|Grothendieck universe axiom]] &#40;the existence of a proper class of [[inaccessible]] cardinals)<br />
* [[inaccessible#Weakly inaccessible cardinal|weakly inaccessible]] cardinal, &#40;strongly) '''[[inaccessible]]''' cardinal<br />
* [[Morse-Kelley set theory|Morse-Kelley]] set theory<br />
* '''[[worldly]]''' cardinal and the [[worldly#Degrees of worldliness|$\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness|hyper-worldly]] cardinal<br />
* the [[Transitive ZFC model#Transitive model universe axiom|transitive model universe axiom]] <br />
* [[transitive ZFC model|transitive model of $\text{ZFC}$]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC|minimal transitive model]]<br />
* '''[[Con ZFC|$\text{Con&#40;ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy|$\text{Con}^\alpha&#40;\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy|iterated consistency hierarchy]]<br />
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br />
<br />
* down to [[the middle attic]]</div>BartekChomhttp://cantorsattic.info/index.php?title=L%C3%A9vy_scheme&diff=4153Lévy scheme2022-05-14T19:37:53Z<p>BartekChom: #REDIRECT Reflecting cardinals</p>
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<div>#REDIRECT [[Reflecting cardinals]]</div>BartekChomhttp://cantorsattic.info/index.php?title=Talk:Indescribable&diff=4144Talk:Indescribable2022-05-14T19:20:01Z<p>BartekChom: /* Strong indescribability */ Can "strong" stay?</p>
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<div>==Strong indescribability==<br />
Thanks for the edit BartekChom! I believe strongly-Q-indescribable is now nonstandard terminology, due to the age of Richter and Aczel's paper some notational choices (including using ω<sub>1</sub> to denote the Church-Kleene ordinal) are not commonly used anymore. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 12:09, 14 May 2022 (PDT)<br />
: Thank you. Can "strong" stay where it is in the source, or should I remove it? [[User:BartekChom|BartekChom]] ([[User talk:BartekChom|talk]]) 12:20, 14 May 2022 (PDT)</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4142Indescribable2022-05-14T19:17:38Z<p>BartekChom: weak is apparently not default</p>
<hr />
<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
<br />
Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
<br />
Here are some other equivalent definitions:<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
<br />
==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
<br />
One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
<br />
===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
<br />
'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
<br />
===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
<br />
Language $\mathcal{L}$ has variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name &#40;individual constant) for each set and a name &#40;relation symbol) for each relation on sets. &#40;§1) ''TODO: complete the definition'' $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)<br />
<br />
We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
<br />
We call $\alpha$ weakly $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
<br />
$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5<!--first part and above-->)<br />
<br />
We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5<!--second part-->)<br />
<br />
We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)<br />
<br />
We call $\alpha$ $Q$-[[reflecting ordinal|reflecting]] on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. &#40;definition 1.7)<!--This is probably not a good place for it, but I cannot organise it better by now.--> With full $\mathcal{L}$ this would yield weak $Q$-indescribability on $X$. (above definition 1.7)<br />
<br />
Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
<br />
==Facts==<br />
<br />
Here are some known facts about indescribability:<br />
<br />
Weak $\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> Strong<sup>&#40;definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite></sup> $\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to weak $\Pi_0^0$-indescribablity on $X$ and to weak $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to weak $\Pi_2^0$-indescribablity on $X$ and to weak $\Pi_0^1$-indescribablity on $X$. Weak $\Pi_n^1$-indescribablity on $X$ is equivalent to weak $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
If $m>2$ or $n>0$, weak $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, weak $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
When $Q$ is $\Pi_m^n$ or $\Sigma_m^n$ for $n>0$, an ordinal is strongly $Q$-indescribable iff it is weakly $Q$-indescribable and strongly inaccessible &#40;therefore strong and weak $Q$-inaccessibility coincide assuming GCH.). &#40;after definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
<br />
[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
<br />
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
<br />
If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
<br />
Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
<br />
$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
<br />
$Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. In particular, ''$Π_3$-reflecting'' or ''2-[[admissible]]'' ordinals can be called ''recursively [[weakly compact]]''. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><cite>Madore2017:OrdinalZoo</cite><br />
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{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4138Admissible2022-05-14T15:10:19Z<p>BartekChom: /* Equivalent definitions */ the</p>
<hr />
<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
<br />
$Π_2$-[[reflecting ordinal]]s are precisely the admissible ordinals $>\omega$. &#40;theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
== Higher admissibility ==<br />
=== Computably inaccessible ordinal ===<br />
<br />
An ordinal $\alpha$ is ''computably [[inaccessible]]'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f&#40;\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
=== Recursively Mahlo ===<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
===2-admissible===<br />
We call $\kappa\in\mathrm{Ad}$ ''2-admissible'' iff every $\xi<\kappa$ such that $\{\xi\}_\kappa$ maps $\kappa$-recursive functions to $\kappa$-recursive functions has a witness &#40;$\alpha\in\mathrm{Ad}$ such that $\xi<\alpha<\kappa$ and $\{\xi\}_\kappa$ maps $\alpha$-recursive functions to $\alpha$-recursive functions). &#40;$\mathrm{Ad}$ is the class of admissible ordinals greater than $\omega$.) ''TODO: complete definition'' &#40;definition 1.15)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissible ordinals are precisely the $Π_3$-[[reflecting ordinal]]s. &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissibility is a recursive analogue of 2-regularity, which is equivalent to [[weakly compact|weak compactness]]. &#40;theorem 1.14)<cite>RichterAczel1974:InductiveDefinitions</cite> ''2-admissible'' ordinals can be called ''recursively weakly compact''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
===$\Sigma_n$-admissible===<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
$\Sigma_n$-admissible ordinals need not necessarily satisfy the $\Sigma_n$-separation schema. For example, the least $\Sigma_2$-admissible ordinal doesn't satisfy $\Sigma_2$-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals &#40;Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] &#40;if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n&#40;L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). &#40;However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n&#40;L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. &#40;Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}&#40;\beta)=\beta$ &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}&#40;\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}&#40;\omega_1^{CK}\times 2)=\omega$. &#40;This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}&#40;\beta)=\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals &#40;M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}&#40;\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}&#40;\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}&#40;\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}&#40;\beta)\textrm{ is a cardinal}"$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}&#40;\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}&#40;\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}&#40;\alpha&#x29;=\Sigma_2\textrm{-proj}&#40;\alpha&#x29;>\Sigma_3\textrm{-proj}&#40;\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $&#40;\Sigma_k\textrm{-proj&#x7d;&#40;\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
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{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4137Reflecting ordinal2022-05-14T15:09:49Z<p>BartekChom: /* Properties */ the</p>
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<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.7</sup><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
==Properties==<br />
$Π_2$-reflecting ordinals are precisely the [[admissible]] ordinals $>\omega$. &#40;theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
''TODO (equivalences, recursively [[Mahlo]])'' &#40;theorem 1.9)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\alpha$ is $Q$-reflecting on $X$ iff $\alpha$ reflects every $Q$-sentence of $\mathcal{L}_p$ on $X$, where $\mathcal{L}_p$ is the sublanguage of $\mathcal{L}$ with relation<br />
symbols only for the primitive recursive relations on sets. ''TODO: complete'' &#40;theorem 1.10)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$Π_3$-reflecting ordinals are precisely 2-[[admissible]] ordinals &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$&#40;+1)$-[[stable]] ordinals are exactly the $Π^1_0$-reflecting &#40;i.e., $Π_n$-reflecting for every $n ∈ ω$<cite>Madore2017:OrdinalZoo</cite>) ordinals &#40;Theorem 1.18). $&#40;{}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals &#40;Theorem 1.19).<cite>RichterAczel1974:InductiveDefinitions</cite><!--Page 18 in the PDF, with label 16--><br />
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{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4136Admissible2022-05-14T15:08:59Z<p>BartekChom: /* Equivalent definitions */ $Π_2$-reflecting ordinals</p>
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<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
<br />
$Π_2$-[[reflecting ordinal]]s are precisely admissible ordinals $>\omega$. &#40;theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
== Higher admissibility ==<br />
=== Computably inaccessible ordinal ===<br />
<br />
An ordinal $\alpha$ is ''computably [[inaccessible]]'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f&#40;\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
=== Recursively Mahlo ===<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
===2-admissible===<br />
We call $\kappa\in\mathrm{Ad}$ ''2-admissible'' iff every $\xi<\kappa$ such that $\{\xi\}_\kappa$ maps $\kappa$-recursive functions to $\kappa$-recursive functions has a witness &#40;$\alpha\in\mathrm{Ad}$ such that $\xi<\alpha<\kappa$ and $\{\xi\}_\kappa$ maps $\alpha$-recursive functions to $\alpha$-recursive functions). &#40;$\mathrm{Ad}$ is the class of admissible ordinals greater than $\omega$.) ''TODO: complete definition'' &#40;definition 1.15)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissible ordinals are precisely the $Π_3$-[[reflecting ordinal]]s. &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissibility is a recursive analogue of 2-regularity, which is equivalent to [[weakly compact|weak compactness]]. &#40;theorem 1.14)<cite>RichterAczel1974:InductiveDefinitions</cite> ''2-admissible'' ordinals can be called ''recursively weakly compact''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
===$\Sigma_n$-admissible===<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
$\Sigma_n$-admissible ordinals need not necessarily satisfy the $\Sigma_n$-separation schema. For example, the least $\Sigma_2$-admissible ordinal doesn't satisfy $\Sigma_2$-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals &#40;Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] &#40;if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n&#40;L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). &#40;However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n&#40;L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. &#40;Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}&#40;\beta)=\beta$ &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}&#40;\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}&#40;\omega_1^{CK}\times 2)=\omega$. &#40;This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}&#40;\beta)=\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals &#40;M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}&#40;\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}&#40;\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}&#40;\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}&#40;\beta)\textrm{ is a cardinal}"$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}&#40;\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}&#40;\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}&#40;\alpha&#x29;=\Sigma_2\textrm{-proj}&#40;\alpha&#x29;>\Sigma_3\textrm{-proj}&#40;\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $&#40;\Sigma_k\textrm{-proj&#x7d;&#40;\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4135Reflecting ordinal2022-05-14T15:07:55Z<p>BartekChom: /* Properties */ properties</p>
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<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.7</sup><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
==Properties==<br />
$Π_2$-reflecting ordinals are precisely [[admissible]] ordinals $>\omega$. &#40;theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
''TODO (equivalences, recursively [[Mahlo]])'' &#40;theorem 1.9)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\alpha$ is $Q$-reflecting on $X$ iff $\alpha$ reflects every $Q$-sentence of $\mathcal{L}_p$ on $X$, where $\mathcal{L}_p$ is the sublanguage of $\mathcal{L}$ with relation<br />
symbols only for the primitive recursive relations on sets. ''TODO: complete'' &#40;theorem 1.10)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$Π_3$-reflecting ordinals are precisely 2-[[admissible]] ordinals &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$&#40;+1)$-[[stable]] ordinals are exactly the $Π^1_0$-reflecting &#40;i.e., $Π_n$-reflecting for every $n ∈ ω$<cite>Madore2017:OrdinalZoo</cite>) ordinals &#40;Theorem 1.18). $&#40;{}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals &#40;Theorem 1.19).<cite>RichterAczel1974:InductiveDefinitions</cite><!--Page 18 in the PDF, with label 16--><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4134Admissible2022-05-14T14:51:22Z<p>BartekChom: 2-admissible</p>
<hr />
<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
== Higher admissibility ==<br />
=== Computably inaccessible ordinal ===<br />
<br />
An ordinal $\alpha$ is ''computably [[inaccessible]]'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f&#40;\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
=== Recursively Mahlo ===<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
===2-admissible===<br />
We call $\kappa\in\mathrm{Ad}$ ''2-admissible'' iff every $\xi<\kappa$ such that $\{\xi\}_\kappa$ maps $\kappa$-recursive functions to $\kappa$-recursive functions has a witness &#40;$\alpha\in\mathrm{Ad}$ such that $\xi<\alpha<\kappa$ and $\{\xi\}_\kappa$ maps $\alpha$-recursive functions to $\alpha$-recursive functions). &#40;$\mathrm{Ad}$ is the class of admissible ordinals greater than $\omega$.) ''TODO: complete definition'' &#40;definition 1.15)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissible ordinals are precisely the $Π_3$-[[reflecting ordinal]]s. &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
2-admissibility is a recursive analogue of 2-regularity, which is equivalent to [[weakly compact|weak compactness]]. &#40;theorem 1.14)<cite>RichterAczel1974:InductiveDefinitions</cite> ''2-admissible'' ordinals can be called ''recursively weakly compact''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
===$\Sigma_n$-admissible===<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
$\Sigma_n$-admissible ordinals need not necessarily satisfy the $\Sigma_n$-separation schema. For example, the least $\Sigma_2$-admissible ordinal doesn't satisfy $\Sigma_2$-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals &#40;Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] &#40;if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n&#40;L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). &#40;However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n&#40;L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. &#40;Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}&#40;\beta)=\beta$ &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}&#40;\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}&#40;\omega_1^{CK}\times 2)=\omega$. &#40;This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}&#40;\beta)=\beta$ &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals &#40;M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}&#40;\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. &#40;K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}&#40;\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}&#40;\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}&#40;\beta)\textrm{ is a cardinal}"$. &#40;W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}&#40;\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}&#40;\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}&#40;\alpha&#x29;=\Sigma_2\textrm{-proj}&#40;\alpha&#x29;>\Sigma_3\textrm{-proj}&#40;\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $&#40;\Sigma_k\textrm{-proj&#x7d;&#40;\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4133Admissible2022-05-14T14:22:42Z<p>BartekChom: /* Higher admissibility */ 2-admissible almost defined</p>
<hr />
<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
== Computably inaccessible ordinal ==<br />
<br />
An ordinal $\alpha$ is ''computably [[inaccessible]]'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f(\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
== Recursively Mahlo and further ==<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
There are also ''recursively [[weakly compact]]'' i.e. ''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-admissible'' ordinals.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
==Higher admissibility==<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
\(\Sigma_n\)-admissible ordinals need not necessarily satisfy the \(\Sigma_n\)-separation schema. For example, the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals (Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] (if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
We call $\kappa\in\mathrm{Ad}$ 2-admissible iff every $\xi<\kappa$ such that $\{\xi\}_\kappa$ maps $\kappa$-recursive functions to $\kappa$-recursive functions has a witness &#40;$\alpha\in\mathrm{Ad}$ such that $\xi<\alpha<\kappa$ and $\{\xi\}_\kappa$ maps $\alpha$-recursive functions to $\alpha$-recursive functions). ($\mathrm{Ad}$ is the class of admissible ordinals greater than $\omega$.) ''TODO: complete definition'' It is a a recursive analogue of 2-regularity. (definition 1.15)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n(L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). (However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n(L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. (Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}(\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}(\omega_1^{CK}\times 2)=\omega$. (This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals (M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}(\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}(\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}(\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}(\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}(\alpha&#x29;=\Sigma_2\textrm{-proj}(\alpha&#x29;>\Sigma_3\textrm{-proj}(\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $(\Sigma_k\textrm{-proj&#x7d;(\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4132Reflecting ordinal2022-05-14T14:05:53Z<p>BartekChom: /* Definition */ from this point of view we have a theorem</p>
<hr />
<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.7</sup><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
==Properties==<br />
''$Π_3$-reflecting'' ordinals are precisely ''2-[[admissible]]'' ordinals &#40;theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''. More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><cite>Madore2017:OrdinalZoo</cite><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4131Indescribable2022-05-14T13:56:12Z<p>BartekChom: /* Facts */ analogy</p>
<hr />
<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
<br />
Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
<br />
Here are some other equivalent definitions:<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
<br />
==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
<br />
One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
<br />
===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
<br />
'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
<br />
===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
<br />
Language $\mathcal{L}$ has variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name &#40;individual constant) for each set and a name &#40;relation symbol) for each relation on sets. &#40;§1) ''TODO: complete the definition'' $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)<br />
<br />
We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
<br />
We call $\alpha$ weakly $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
<br />
$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5<!--first part and above-->)<br />
<br />
We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5<!--second part-->)<br />
<br />
We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)<br />
<br />
We call $\alpha$ $Q$-[[reflecting ordinal|reflecting]] on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. &#40;definition 1.7)<!--This is probably not a good place for it, but I cannot organise it better by now.--> With full $\mathcal{L}$ this would yield &#40;weak) $Q$-indescribability on $X$. (above definition 1.7)<br />
<br />
Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
<br />
==Facts==<br />
<br />
Here are some known facts about indescribability:<br />
<br />
Weak $\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> Strong<sup>&#40;definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite></sup> $\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to weak $\Pi_0^0$-indescribablity on $X$ and to weak $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to weak $\Pi_2^0$-indescribablity on $X$ and to weak $\Pi_0^1$-indescribablity on $X$. Weak $\Pi_n^1$-indescribablity on $X$ is equivalent to weak $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
If $m>2$ or $n>0$, weak $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, weak $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
When $Q$ is $\Pi_m^n$ or $\Sigma_m^n$ for $n>0$, an ordinal is strongly $Q$-indescribable iff it is weakly $Q$-indescribable and strongly inaccessible &#40;therefore strong and weak $Q$-inaccessibility coincide assuming GCH.). &#40;after definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
<br />
[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
<br />
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
<br />
If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
<br />
Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
<br />
$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
<br />
$Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. In particular, ''$Π_3$-reflecting'' or ''2-[[admissible]]'' ordinals can be called ''recursively [[weakly compact]]''. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><cite>Madore2017:OrdinalZoo</cite><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Weakly_compact&diff=4130Weakly compact2022-05-14T13:49:17Z<p>BartekChom: /* $\Sigma_n$-weakly compact etc. */ recursive analogue</p>
<hr />
<div>{{DISPLAYTITLE: Weakly compact cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
Weakly compact cardinals lie at the focal point of a number<br />
of diverse concepts in infinite combinatorics, admitting various characterizations in terms of these concepts. If $\kappa^{{<}\kappa} = \kappa$, then the following are equivalent: <br />
<br />
:; Weak compactness : A cardinal $\kappa$ is weakly compact if and only if it is [[uncountable]] and every $\kappa$-satisfiable theory in an [[Infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] language of size at most $\kappa$ is satisfiable.<br />
:; Extension property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset V_\kappa$, there is a transitive structure $W$ properly extending $V_\kappa$ and $A^*\subset W$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.<br />
:; Tree property : A cardinal $\kappa$ is weakly compact if and only if it is [[inaccessible]] and has the [[tree property]].<br />
:; Filter property : A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a set containing at most $\kappa$-many subsets of $\kappa$, then there is a $\kappa$-[[filter|complete nonprincipal filter]] $F$ measuring every set in $M$.<br />
:; Weak embedding property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with $\kappa\in M$ and a transitive set $N$ with an [[elementary embedding|embedding]] $j:M\to N$ with [[critical point]] $\kappa$.<br />
:; Embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a transitive set $N$ and an embedding $j:M\to N$ with critical point $\kappa$.<br />
:; Normal embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$, such that $N=\{\ j(f)(\kappa)\mid f\in M\ \}$.<br />
;; Hauser embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.<br />
:; Partition property : A cardinal $\kappa$ is weakly compact if and only if the [[partition property]] $\kappa\to(\kappa)^2_2$ holds.<br />
:; Indescribability property : A cardinal $\kappa$ is weakly compact if and only if it is $\Pi_1^1$-[[indescribable]].<br />
:; Skolem Property : A cardinal $\kappa$ is weakly compact if and only if $\kappa$ is inaccessible and every $\kappa$-unboundedly satisfiable $\mathcal{L}_{\kappa,\kappa}$-theory $T$ of size at most $\kappa$ has a model of size at least $\kappa$. A theory $T$ is $\kappa$-unboundedly satisfiable if and only if for any $\lambda<\kappa$, there exists a model $\mathcal{M}\models T$ with $\lambda\leq|M|<\kappa$. For more info see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937#309937 here].<br />
<br />
Weakly compact cardinals first arose<br />
in connection with (and were named for) the question of<br />
whether certain [[Infinitary logic|infinitary logics]] satisfy the compactness<br />
theorem of first order logic. Specifically, in a language<br />
with a signature consisting, as in the first order context,<br />
of a set of constant, finitary function and relation<br />
symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$<br />
formulas by closing the collection of formulas under<br />
infinitary conjunctions<br />
$\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions<br />
$\vee_{\alpha<\delta}\varphi_\alpha$ of any size<br />
$\delta<\kappa$, as well as infinitary quantification<br />
$\exists\vec x$ and $\forall\vec x$ over blocks of<br />
variables $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less<br />
than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics.<br />
A theory is ''$\theta$-satisfiable'' if every subtheory<br />
consisting of fewer than $\theta$ many sentences of it is<br />
satisfiable. First order logic is precisely<br />
$L_{\omega,\omega}$, and the classical Compactness theorem<br />
asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$<br />
theory is satisfiable. A uncountable cardinal $\kappa$ is<br />
''[[strongly compact]]'' if every $\kappa$-satisfiable<br />
$\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal<br />
$\kappa$ is ''weakly compact'' if every<br />
$\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a<br />
language having at most $\kappa$ many constant, function<br />
and relation symbols, is satisfiable.<br />
<br />
Next, for any cardinal $\kappa$, a ''$\kappa$-tree'' is a<br />
tree of height $\kappa$, all of whose levels have size less<br />
than $\kappa$. More specifically, $T$ is a ''tree'' if<br />
$T$ is a partial order such that the predecessors of any<br />
node in $T$ are well ordered. The $\alpha^{\rm th}$ level of a<br />
tree $T$, denoted $T_\alpha$, consists of the nodes whose<br />
predecessors have order type exactly $\alpha$, and these<br />
nodes are also said to have ''height'' $\alpha$. The height of the tree $T$ is the first $\alpha$ for which $T$<br />
has no nodes of height $\alpha$. A ""$\kappa$-branch""<br />
through a tree $T$ is a maximal linearly ordered subset of<br />
$T$ of order type $\kappa$. Such a branch selects exactly<br />
one node from each level, in a linearly ordered manner. The<br />
set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree<br />
is an ''Aronszajn'' tree if it has no $\kappa$-branches.<br />
A cardinal $\kappa$ has the ''tree property'' if every<br />
$\kappa$-tree has a $\kappa$-branch.<br />
<br />
A transitive set $M$ is a $\kappa$-model of set theory if<br />
$|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$,<br />
the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement). <br />
For any<br />
infinite cardinal $\kappa$ we have<br />
that $H_{\kappa^+}$ models ZFC$^-$, and further, if<br />
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is<br />
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed<br />
into such an $M$. If $\kappa^{\lt\kappa}=\kappa$, one can use<br />
the downward L&ouml;wenheim-Skolem theorem to find such $M$<br />
with $M^{\lt\kappa}\subset M$. So in this case there are abundant<br />
$\kappa$-models of set theory (and conversely, if there is<br />
a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$).<br />
<br />
The partition property $\kappa\to(\lambda)^n_\gamma$<br />
asserts that for every function $F:[\kappa]^n\to\gamma$<br />
there is $H\subset\kappa$ with $|H|=\lambda$ such that<br />
$F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as<br />
coloring the $n$-tuples, the partition property asserts the<br />
existence of a ''monochromatic'' set $H$, since all<br />
tuples from $H$ get the same color. The partition property<br />
$\kappa\to(\kappa)^2_2$ asserts that every partition of<br />
$[\kappa]^2$ into two sets admits a set $H\subset\kappa$ of<br />
size $\kappa$ such that $[H]^2$ lies on one side of the<br />
partition. When defining $F:[\kappa]^n\to\gamma$, we define<br />
$F(\alpha_1,\ldots,\alpha_n)$ only when<br />
$\alpha_1<\cdots<\alpha_n$.<br />
<br />
== Weakly compact cardinals and the constructible universe ==<br />
<br />
Every weakly compact cardinal is weakly compact in [[Constructible universe|$L$]]. <cite>Jech2003:SetTheory</cite><br />
<br />
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. <br />
<br />
<br />
== Weakly compact cardinals and forcing ==<br />
<br />
* Weakly compact cardinals are invariant under small forcing. [http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf]<br />
* Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions {{Citation needed}}.<br />
* If $\kappa$ is weakly compact, there is a forcing extension in which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa^+$ {{Citation needed}}. <br />
* If the existence of weakly compact cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not weakly compact, but becomes weakly compact in a forcing extension <CITE>Kunen1978:SaturatedIdeals</CITE>.<br />
<br />
== Indestructibility of a weakly compact cardinal ==<br />
''To expand using [https://arxiv.org/abs/math/9907046]''<br />
<br />
== Relations with other large cardinals == <br />
<br />
* Every weakly compact cardinal is [[inaccessible]], [[Mahlo]], hyper-Mahlo, hyper-hyper-Mahlo and more. <br />
* [[Measurable]] cardinals, [[Ramsey]] cardinals, and [[indescribable|totally indescribable]] cardinals are all weakly compact and a stationary limit of weakly compact cardinals.<br />
* Assuming the consistency of a [[strongly unfoldable]] cardinal with ZFC, it is also consistent for the least weakly compact cardinal to be the least [[unfoldable]] cardinal. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If GCH holds, then the least weakly compact cardinal is not [[weakly measurable]]. However, if there is a [[measurable]] cardinal, then it is consistent for the least weakly compact cardinal to be weakly measurable. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If it is consistent for there to be a [[nearly supercompact]], then it is consistent for the least weakly compact cardinal to be nearly supercompact. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
* For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-[[Ramsey]]. <cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
==$\Sigma_n$-weakly compact etc.==<br />
An inaccessible cardinal $κ$ is $Σ_n$-weakly compact iff it reflects $Π_1^1$ sentences with $Σ_n$-predicates, i.e. for every $R ⊆ V_κ$ which is definable by a $Σ_n$ formula &#40;with parameters) over $V_κ$ and every $Π_1^1$ sentence $Φ$, if $\langle V_κ , ∈, R \rangle \models Φ$ then there is $α < κ$ &#40;equivalently, unboundedly-many $α < κ$) such that $\langle V_α , ∈, R ∩ V_α \rangle \models Φ$. Analogously for $Π_n$ and $∆_n$. $κ$ is $Σ_ω$-weakly compact iff it is $Σ_n$-weakly compact for all $n < ω$.<br />
<br />
$κ$ is $Σ_n$-weakly compact $\iff$ $κ$ is $Π_n$-weakly compact $\iff$ $κ$ is $∆_{n+1}$-weakly compact $\iff$ For every $Π_1^1$ formula $Φ&#40;x_0 , ..., x_k)$ in the language of set theory and every<br />
$a_0 , ..., a_k ∈ V_κ$, if $V κ \models Φ&#40;a_0 , ..., a_k )$, then there is $λ ∈ I_n := \{λ < κ : λ$ is inaccessible and $V_λ \preccurlyeq_n V_κ\}$ such that $V_λ \models Φ&#40;a_0 , ..., a_k)$.<br />
<br />
In <cite>Bosch2006:SmallDefinablyLargeCardinals</cite> it is shown that every $Σ_ω$-w.c. cardinal is $Σ_ω$-[[Mahlo]] and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.<br />
<br />
These properties are connected with [[axioms of generic absoluteness]]. For example:<br />
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.<br />
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.<br />
<br />
This section from<cite>Leshem2000:OCDefinableTreeProperty</cite><cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
<br />
==Recursive analogue==<br />
''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-[[admissible]]'' ordinals are analogous to weakly compact &#40;$Π_1^1$-indescribable) cardinals and can be called ''recursively weakly compact''<cite>Madore2017:OrdinalZoo</cite><cite>RichterAczel1974:InductiveDefinitions</cite><sup>after definition 1.12</sup><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4129Admissible2022-05-14T13:41:37Z<p>BartekChom: /* Computably inaccessible ordinal */ [[]]</p>
<hr />
<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
== Computably inaccessible ordinal ==<br />
<br />
An ordinal $\alpha$ is ''computably [[inaccessible]]'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f(\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
== Recursively Mahlo and further ==<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
There are also ''recursively [[weakly compact]]'' i.e. ''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-admissible'' ordinals.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
==Higher admissibility==<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
\(\Sigma_n\)-admissible ordinals need not necessarily satisfy the \(\Sigma_n\)-separation schema. For example, the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals (Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] (if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n(L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). (However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n(L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. (Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}(\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}(\omega_1^{CK}\times 2)=\omega$. (This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals (M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}(\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}(\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}(\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}(\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}(\alpha&#x29;=\Sigma_2\textrm{-proj}(\alpha&#x29;>\Sigma_3\textrm{-proj}(\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $(\Sigma_k\textrm{-proj&#x7d;(\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Admissible&diff=4128Admissible2022-05-14T13:40:57Z<p>BartekChom: /* Recursively Mahlo and further */ RichterAczel1974:InductiveDefinitions</p>
<hr />
<div>[[Category:Lower attic]]<br />
<br />
{{stub}}<br />
<br />
An ordinal $\gamma$ is ''admissible'' if the $L_\gamma$ level of the [[constructible universe]] satisfies the [[Kripke-Platek]] axioms of set theory. The term was coined by Richard Platek in 1966.<!--Source: Barwise, "Part C: α-Recursion--><br />
<br />
The smallest admissible ordinal is often considered to be [[omega|$\omega$]], the least infinite ordinal. However, some authors<!--Such as Christoph Duchhardt--> include Infinity in the KP axioms, in which case [[Church-Kleene|$\omega_1^{CK}$]],<cite>Madore2017:OrdinalZoo</cite> the least non-computable ordinal, is the least admissible. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$. <br />
<br />
The smallest limit of admissible ordinals, $\omega_\omega^{CK}$, is not admissible.<cite>Madore2017:OrdinalZoo</cite><br />
==Equivalent definitions==<br />
The following properties are also equivalent to admissibility:<br />
*<br />
*<br />
== Computably inaccessible ordinal ==<br />
<br />
An ordinal $\alpha$ is ''computably inaccessible'', also known as ''recursively inaccessible'', if it is admissible and a limit of admissible ordinals.<cite>Madore2017:OrdinalZoo</cite> If <math>f</math> enumerates admissible ordinals, recursively inaccessible ordinals are exactly the ordinals <math>\alpha</math> where <math>\alpha=f(\alpha)</math>.<!--Barwise, Admissible Sets and Structures (p.176)--><br />
<br />
== Recursively Mahlo and further ==<br />
An ordinal $α$ is ''recursively [[Mahlo]]'' iff for any [https://en.wikipedia.org/wiki/Alpha_recursion_theory $α$-recursive function] $f : α → α$ there is an admissible $β < α$ closed under $f$.<cite>Madore2017:OrdinalZoo</cite><br />
<br />
There are also ''recursively [[weakly compact]]'' i.e. ''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-admissible'' ordinals.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. &#40;after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
==Higher admissibility==<br />
[[File:AdmissibilityImplicationDiagram.png|thumb|Some implications between admissibility-related conditions.]]<br />
<br />
Admissibility has been extended using stronger collection axioms. One common formulation is that an ordinal $\alpha$ is $\Sigma_n$'''-admissible''' if $L_\alpha\vDash\textrm{RST}\cup\Sigma_n\textrm{-collection}$<!--Kranakis citation-->, where $\textrm{RST}$ denotes rudimentary set theory, i.e. Kripke-Platek set theory without the $\Sigma_0$-collection axiom<!--Aczel citation here-->.<br />
<br />
\(\Sigma_n\)-admissible ordinals need not necessarily satisfy the \(\Sigma_n\)-separation schema. For example, the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation.<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α. Cf. https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf#page=19--><br />
<br />
Here are some properties of $\Sigma_n$-admissibility:<br />
*$\Sigma_1$-admissibility is equivalent to $\Sigma_0$-admissibility.<br />
*For $n>1$, $\Sigma_n$-admissibility can be couched in terms of [[Reflecting ordinal|reflection]] onto sets of [[stable]] ordinals (Kranakis), and the smallest $\Sigma_n$-admissible ordinal is greater then the smallest [[nonprojectible]] ordinal and weaker variants of [[stable]] ordinals but smaller than the height of the [[Transitive ZFC model|minimal model of ZFC]] (if it exists).<cite>Madore2017:OrdinalZoo</cite><br />
<br />
==Cofinality and projectum==<br />
Two concepts used in the study of admissible ordinals are $\Sigma_1$-cofinality and $\Sigma_1$-projecta.<br />
*The $\Sigma_1$-cofinality of $\beta$ is the least $\xi$ such that there exists a $\Sigma_1$-definable function mapping $\xi$ cofinally into $\beta$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*The $\Sigma_n$-projectum of $\beta$ is equal to<!--but not defined as--> the least $\delta$ such that some $\Sigma_n(L_\alpha)$-definable function maps a subset of $<!--\omega-->\delta$ onto $L_\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972). (However note that when using the Jensen hierarchy instead of the hierarchy $L$, as the original source does, that behavior may change)<br />
**Alternatively, when $n=1$, the $\Sigma_1$-projectum of $\beta$ has been given as the least $\gamma\le\beta$ such that a $\beta$-recursive one-to-one function $f:\beta\rightarrow\gamma$ exists. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<nowiki>[</nowiki>Barwise<!-- "Part C: α-Recursion-->, p.157] This is claimed to extend to $n>1$ in [https://arxiv.org/pdf/math/9609203.pdf]<br />
**Alternatively, the $\Sigma_n$-projectum of $\alpha$ is the smallest $\rho$ such that there exists a $\Sigma_n(L_\alpha)$ function $f$ with $f^{\prime\prime}L_\rho=L_\alpha$.<cite>Jech2003:SetTheory</cite><sup>p.549</sup><br />
<br />
$\Delta_n$-projecta are similar to $\Sigma_n$-projecta, except that its behavior lacks the involvement of a bounded subset of $<!--\omega-->\delta$, employing just the ordinal $<!--\omega-->\delta$ instead. (Compare Σ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=50], Δ<sub>n</sub>: [https://core.ac.uk/download/pdf/30905237.pdf#page=52])<br />
===Properties===<br />
*$\beta$ is admissible iff $\Sigma_1\textrm{-cof}(\beta)=\beta$ (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976).<br />
**Note that although admissibility is considered to be "recursive regularity"<cite>Arai97:P</cite><sup>p.4</sup>, $\Sigma_1\textrm{-cof}$ behaves differently with respect to admissibles than $\textrm{cof}$ does with respect to regular cardinals. For example, $\textrm{cof}(\omega_1\times 2)=\omega_1$, however $\Sigma_1\textrm{-cof}(\omega_1^{CK}\times 2)=\omega$. (This is because there's a one-to-one map $f:\omega_1^{CK}\rightarrow\omega$ that's $\omega_1^{CK}$-recursive<!--Barwise, "Part C: α-Recursion (p.157)-->,therefore also $\omega_1^{CK}\times 2$-recursive)<br />
*$\beta$ is [[Stable#Variants|nonprojectible]] iff $\Sigma_1\textrm{-proj}(\beta)=\beta$ (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=39 An introduction to the fine structure of the constructible hierarchy], 1972).<br />
**For the first alternative definition of the $\Sigma_1$-projectum, compare to Rathjen's description of nonprojectible ordinals (M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf#page=18 The Art of Ordinal Analysis]).<br />
*A more fine but extendable result, if we assume $n>1$ and $\omega\beta=\beta$, then $\Sigma_&#x6e;\textrm{-proj}(\beta)>\omega$ iff $\beta$ begins a $\Sigma_n$-gap. (K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf#page=50 An introduction to the fine structure of the constructible hierarchy], 1972)<!--Patterns of Projecta also mentions this result via their definition of projectum, but it's the same theorem also proven by Jensen-->. Similarly, if $\Delta_n\textrm{-proj}(\beta)>\omega$ then $\beta$ begins a $\Delta_n$-gap.<br />
*When $\beta$ is admissible{{citation needed}}<!--I think because it requires β-recursion theory, which implicitly assumes β admissible-->, $L_\beta\vDash``\Sigma_1\textrm{-cof}(\beta)\textrm{ is a cardinal}"$ and $L_\beta\vDash``\Sigma_1\textrm{-proj}(\beta)\textrm{ is a cardinal}"$. (W. Maass, [https://igi-web.tugraz.at/PDF/4.pdf#page=3 Inadmissibility, tame R.E. sets and the admissible collapse], 1976)<br />
*Applying a result from [[Heights_of_models#ZFC_without_the_powerset_axiom|here]], $L_&#x7b;\Sigma_1\textrm{-cof}(\beta&#x29;}$[[Stable|$\prec_{\Sigma_1}$]]$L_\beta$ and $L_&#x7b;\Sigma_1\textrm{-proj}(\beta&#x29;}\prec_{\Sigma_1}L_\beta$.<br />
<!--Σ_n-projecta of a constant ordinal may decrease as n increases. [https://math.stackexchange.com/questions/1635440/an-exercise-in-fine-structure-of-constructible-universe-concerning-projectum-pat]--><br />
<br />
===Patterns===<br />
Sometimes, some unintuitive patterns arise in projecta of an ordinal, such as $\Sigma_1\textrm{-proj}(\alpha&#x29;=\Sigma_2\textrm{-proj}(\alpha&#x29;>\Sigma_3\textrm{-proj}(\alpha&#x29;$&#x2e; In fact, for any binary string, there exists some ordinal $\alpha$ whose sequence $(\Sigma_k\textrm{-proj&#x7d;(\alpha&#x29;)_{0\le k\le n}$ has pairwise comparisons $>$, $=$ each determined by that string. <sup>citation needed</sup><!--https://www.jstor.org/stable/2273621? https://mathoverflow.net/questions/67933/sequences-of-projecta-in-the-constructible-hierarchy?--><br />
<br />
{{references}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4127Indescribable2022-05-14T13:22:08Z<p>BartekChom: /* Indescribable on a set */ it seems that in modern terminology strong indescribability is default</p>
<hr />
<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
<br />
Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
<br />
Here are some other equivalent definitions:<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
<br />
==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
<br />
One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
<br />
===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
<br />
'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
<br />
===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
<br />
Language $\mathcal{L}$ has variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name &#40;individual constant) for each set and a name &#40;relation symbol) for each relation on sets. &#40;§1) ''TODO: complete the definition'' $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)<br />
<br />
We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
<br />
We call $\alpha$ weakly $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
<br />
$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5<!--first part and above-->)<br />
<br />
We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5<!--second part-->)<br />
<br />
We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)<br />
<br />
We call $\alpha$ $Q$-[[reflecting ordinal|reflecting]] on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. &#40;definition 1.7)<!--This is probably not a good place for it, but I cannot organise it better by now.--> With full $\mathcal{L}$ this would yield &#40;weak) $Q$-indescribability on $X$. (above definition 1.7)<br />
<br />
Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
<br />
==Facts==<br />
<br />
Here are some known facts about indescribability:<br />
<br />
Weak $\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> Strong<sup>&#40;definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite></sup> $\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to weak $\Pi_0^0$-indescribablity on $X$ and to weak $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to weak $\Pi_2^0$-indescribablity on $X$ and to weak $\Pi_0^1$-indescribablity on $X$. Weak $\Pi_n^1$-indescribablity on $X$ is equivalent to weak $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
If $m>2$ or $n>0$, weak $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, weak $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
When $Q$ is $\Pi_m^n$ or $\Sigma_m^n$ for $n>0$, an ordinal is strongly $Q$-indescribable iff it is weakly $Q$-indescribable and strongly inaccessible &#40;therefore strong and weak $Q$-inaccessibility coincide assuming GCH.). &#40;after definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
<br />
[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
<br />
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
<br />
If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
<br />
Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
<br />
$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4126Indescribable2022-05-14T13:18:41Z<p>BartekChom: /* Facts */ weakly and strongly indescribable</p>
<hr />
<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
<br />
Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
<br />
Here are some other equivalent definitions:<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
<br />
==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
<br />
One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
<br />
===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
<br />
'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
<br />
===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
<br />
Language $\mathcal{L}$ has variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name &#40;individual constant) for each set and a name &#40;relation symbol) for each relation on sets. &#40;§1) ''TODO: complete the definition'' $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)<br />
<br />
We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
<br />
We call $\alpha$ &#40;weakly) $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
<br />
$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5<!--first part and above-->)<br />
<br />
We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5<!--second part-->)<br />
<br />
We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)<br />
<br />
We call $\alpha$ $Q$-[[reflecting ordinal|reflecting]] on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. &#40;definition 1.7)<!--This is probably not a good place for it, but I cannot organise it better by now.--> With full $\mathcal{L}$ this would yield &#40;weak) $Q$-indescribability on $X$. (above definition 1.7)<br />
<br />
Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
<br />
==Facts==<br />
<br />
Here are some known facts about indescribability:<br />
<br />
Weak $\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> Strong<sup>&#40;definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite></sup> $\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to weak $\Pi_0^0$-indescribablity on $X$ and to weak $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to weak $\Pi_2^0$-indescribablity on $X$ and to weak $\Pi_0^1$-indescribablity on $X$. Weak $\Pi_n^1$-indescribablity on $X$ is equivalent to weak $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
If $m>2$ or $n>0$, weak $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, weak $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
When $Q$ is $\Pi_m^n$ or $\Sigma_m^n$ for $n>0$, an ordinal is strongly $Q$-indescribable iff it is weakly $Q$-indescribable and strongly inaccessible &#40;therefore strong and weak $Q$-inaccessibility coincide assuming GCH.). &#40;after definition 1.5)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
<br />
[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
<br />
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
<br />
If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
<br />
Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
<br />
$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4125Indescribable2022-05-14T08:59:34Z<p>BartekChom: /* Indescribable on a set */ strongly indescribable, reflecting ordinal</p>
<hr />
<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
<br />
Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
<br />
Here are some other equivalent definitions:<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
<br />
*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
<br />
==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
<br />
One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
<br />
===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
<br />
'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
<br />
$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
<br />
There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
<br />
===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
<br />
Language $\mathcal{L}$ has variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.), a name &#40;individual constant) for each set and a name &#40;relation symbol) for each relation on sets. &#40;§1) ''TODO: complete the definition'' $\mathcal{L}_\in$ is the sublanguage of $\mathcal{L}$ allowing only $\in$ as a relation symbol. (above definition 1.7)<br />
<br />
We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
<br />
We call $\alpha$ &#40;weakly) $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
<br />
$R(\alpha)=\bigcup_{\beta<\alpha} \mathcal{P}(R(\beta))$ for an ordinal $\alpha$. We say that $R(\alpha)$ reflects $\varphi$ on $X$ iff $R(\alpha) \models \varphi \implies \exists_{\beta \in X \cap \alpha} R(\beta) \models \phi$. (definition 1.5<!--first part and above-->)<br />
<br />
We call $\alpha$ strongly $Q$-indescribable on $X$ iff $R(\alpha)$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.5<!--second part-->)<br />
<br />
We say that $L_\alpha$ reflects $\varphi$ on $X$ iff $L_\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} L_\beta \models \phi$. (definition 1.6)<br />
<br />
We call $\alpha$ $Q$-[[reflecting ordinal|reflecting]] on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}_\in$. &#40;definition 1.7)<!--This is probably not a good place for it, but I cannot organise it better by now.--> With full $\mathcal{L}$ this would yield &#40;weak) $Q$-indescribability on $X$. (above definition 1.7)<br />
<br />
Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
<br />
==Facts==<br />
<br />
Here are some known facts about indescribability:<br />
<br />
<!--$\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> ...... a nonequivalent definition? ......--><br />
$\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to $\Pi_0^0$-indescribablity on $X$ and to $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to $\Pi_2^0$-indescribablity on $X$ and to $\Pi_0^1$-indescribablity on $X$. $\Pi_n^1$-indescribablity on $X$ is equivalent to $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
If $m>2$ or $n>0$, $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
<br />
[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
<br />
Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
<br />
Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
<br />
Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
<br />
If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
<br />
Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
<br />
$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4124Reflecting ordinal2022-05-14T08:43:18Z<p>BartekChom: definition 1.7</p>
<hr />
<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.7</sup><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Library&diff=4123Library2022-05-14T08:40:39Z<p>BartekChom: /* Library holdings */ -:P!</p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}[[Category:Cantor's Attic]]<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
Welcome to the library, our central repository for references cited here on Cantor's attic.<br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
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<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0460120 &#40;57 \#116)},<br />
MRREVIEWER = {Thomas J. Jech}<br />
}<br />
<br />
#Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod bibtex=@article {Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod,<br />
AUTHOR = {Apter, Arthur W.},<br />
TITLE = {Some applications of Sargsyan’s equiconsistency method},<br />
JOURNAL = {Fund. Math.},<br />
VOLUME = {216},<br />
PAGES = {207--222},<br />
}<br />
<br />
#Arai97:P bibtex=@paper{Arai97:P,<br />
title={A sneak preview of proof theory of ordinals},<br />
author={Arai, Toshiyasu},<br />
url={https://www.arxiv.org/abs/1102.0596v1},<br />
year={1997}<br />
}<br />
<br />
#Arai2019:FirstOrderReflection bibtex=@paper{Arai2019:FirstOrderReflection,<br />
TITLE = {A simplified ordinal analysis of first-order reflection},<br />
AUTHOR = {Arai, Toshiyasu},<br />
URL = {https://arxiv.org/abs/1907.07611v1},<br />
YEAR = {2019}<br />
}<br />
<br />
#Baaz2011:Kurt bibtex=@book{Baaz2011:Kurt,<br />
title={Kurt Gödel and the Foundations of Mathematics: Horizons of Truth},<br />
author={Baaz, M. and Papadimitriou, C.H. and Putnam, H.W. and Scott, D.S. and Harper, C.L.},<br />
isbn={9781139498432},<br />
url={https://books.google.pl/books?id=Tg0WXU5\_8EgC},<br />
year={2011},<br />
publisher={Cambridge University Press}<br />
}<br />
<br />
#Bagaria2002:AxiomsOfGenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {Axioms of generic absoluteness},<br />
JOURNAL = {Logic Colloquium 2002},<br />
BOOKTITLE = {Logic Colloquium '02: Lecture Notes in Logic 27},<br />
YEAR = {2006},<br />
DOI = {10.1201/9781439865903},<br />
ISBN = {9780429065262},<br />
URL = {https://www.academia.edu/2561575/AXIOMS_OF_GENERIC_ABSOLUTENESS},<br />
}<br />
<br />
#BagariaBosch2004:PFESolovay bibtex=@article {BagariaBosch2004:PFESolovay,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Proper forcing extensions and Solovay models},<br />
JOURNAL = {Archive for Mathematical Logic},<br />
YEAR = {2004},<br />
DOI = {10.1007/s00153-003-0210-2},<br />
URL = {https://www.academia.edu/2561570/Proper_forcing_extensions_and_Solovay_models},<br />
}<br />
<br />
#BagariaBosch2007:GenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Generic absoluteness under projective forcing},<br />
JOURNAL = {Fundamenta Mathematicae},<br />
YEAR = {2007},<br />
VOLUME = {194},<br />
PAGES = {95-120},<br />
DOI = {10.4064/fm194-2-1},<br />
}<br />
<br />
#Bagaria2012:CnCardinals bibtex=@article{Bagaria2012:CnCardinals,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {$C^{&#40;n)}$-cardinals},<br />
journal = {Archive for Mathematical Logic},<br />
YEAR = {2012},<br />
volume = {51},<br />
number = {3--4},<br />
pages = {213--240},<br />
DOI = {10.1007/s00153-011-0261-8},<br />
URL = {http://www.mittag-leffler.se/sites/default/files/IML-0910f-26.pdf},<br />
eprint = {1908.09664}<br />
}<br />
<br />
#BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@article{BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br />
AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosický, Jiří},<br />
TITLE = {Definable orthogonality classes in accessible categories are small},<br />
journal = {Journal of the European Mathematical Society},<br />
volume = {17},<br />
number = {3},<br />
pages = {549--589},<br />
eprint = {1101.2792}<br />
}<br />
<br />
#BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible bibtex=@article{BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible,<br />
author = {Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi},<br />
title = {Superstrong and other large cardinals are never Laver indestructible},<br />
eprint = {1307.3486},<br />
year = {2013},<br />
journal = {Archive for Mathematical Logic},<br />
volume = {55},<br />
number = {1-2},<br />
pages = {19--35},<br />
url = {http://jdh.hamkins.org/superstrong-never-indestructible/},<br />
doi = {10.1007/s00153-015-0458-3}<br />
}<br />
<br />
#Bagaria2017:LargeCardinalsBeyondChoice bibtex=@article{Bagaria2017:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2017},<br />
url = {https://events.math.unipd.it/aila2017/sites/default/files/BAGARIA.pdf}<br />
}<br />
<br />
#BagariaGitmanSchindler2017:VopenkaPrinciple bibtex=@ARTICLE{BagariaGitmanSchindler2017:VopenkaPrinciple,<br />
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},<br />
TITLE = {Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
NUMBER = {1-2},<br />
PAGES = {1--20},<br />
ISSN = {0933-5846},<br />
MRCLASS = {03E35 &#40;03E55 03E57)},<br />
MRNUMBER = {3598793},<br />
DOI = {10.1007/s00153-016-0511-x},<br />
URL = {https://victoriagitman.github.io/publications/2016/02/10/generic-vopenkas-principle-remarkable-cardinals-and-the-weak-proper-forcing-axiom.html}<br />
}<br />
<br />
#BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice bibtex=@article{BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan and Koellner, Peter and Woodin, W. Hugh},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2019},<br />
journal = {Bulletin of Symbolic Logic},<br />
volume = {25},<br />
number = {3},<br />
pages = {283--318},<br />
url = {https://par.nsf.gov/servlets/purl/10149501},<br />
doi = {10.1017/bsl.2019.28}<br />
}<br />
<br />
#Baumgartner1975:Ineffability bibtex=@incollection{Baumgartner1975:Ineffability,<br />
AUTHOR = {Baumgartner, James},<br />
TITLE = {Ineffability properties of cardinals. I},<br />
BOOKTITLE = {Infinite and finite sets &#40;Colloq., Keszthely, 1973; dedicated to P. Erd&#337;s on his 60th birthday), Vol. I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0384553 &#40;52 \#5427)},<br />
MRREVIEWER = {John K. Truss}<br />
}<br />
<br />
#Blass2010:CardinalCharacteristicsHandbook bibtex=@article{Blass2010:CardinalCharacteristicsHandbook,<br />
author = {Blass, Andreas},<br />
title = {Chapter 6: Cardinal characteristics of the continuum},<br />
journal = {Handbook of Set Theory},<br />
editor = {Foreman, Mathew; Kanamori, Akihiro},<br />
year = {2010},<br />
isbn = {1402048432},<br />
publisher = {Springer},<br />
url = {http://www.math.lsa.umich.edu/~ablass/hbk.pdf},<br />
}<br />
<br />
#Blass1976:ExactFunctors bibtex=@article{Blass1976:ExactFunctors,<br />
author = "Blass, Andreas",<br />
fjournal = "Pacific Journal of Mathematics",<br />
journal = "Pacific J. Math.",<br />
number = "2",<br />
pages = "335--346",<br />
publisher = "Pacific Journal of Mathematics, A Non-profit Corporation",<br />
title = "Exact functors and measurable cardinals.",<br />
url = "https://projecteuclid.org:443/euclid.pjm/1102867389",<br />
volume = "63",<br />
year = "1976"<br />
}<br />
<br />
#Boney2017:ModelTheoreticCharacterizations bibtex=@article{BBoney2017:ModelTheoreticCharacterizations,<br />
author = {Boney, Will},<br />
title = {Model Theoretic Characterizations of Large Cardinals},\<br />
year = {2017},<br />
eprint = {1708.07561},<br />
}<br />
<br />
#Bosch2006:SmallDefinablyLargeCardinals bibtex=@article {Bosch2006:SmallDefinablyLargeCardinals,<br />
AUTHOR = {Bosch, Roger},<br />
TITLE = {Small Definably-large Cardinals},<br />
JOURNAL = {Set Theory. Trends in Mathematics},<br />
YEAR = {2006},<br />
PAGES = {55-82},<br />
DOI = {10.1007/3-7643-7692-9_3},<br />
ISBN = {978-3-7643-7692-5},<br />
}<br />
<br />
#Cantor1915:ContributionsFoundingTransfinite bibtex=@book{Cantor1915:ContributionsFoundingTransfinite, <br />
author = {Cantor, Georg}, <br />
title = {Contributions to the Founding of the Theory of Transfinite Numbers}, <br />
editor = {Jourdain, Philip},<br />
note = {Original year was 1915}, <br />
publisher = {Dover}, <br />
address = {New York}, <br />
year = {1955}, <br />
isbn = {978-0-486-60045-1},<br />
url = {http://www.archive.org/details/contributionstot003626mbp},<br />
}<br />
<br />
#Carmody2015:ForceToChangeLargeCardinalStrength bibtex=@article{Carmody2015:ForceToChangeLargeCardinalStrength, <br />
author = {Carmody, Erin Kathryn}, <br />
title = {Force to change large cardinal strength}, <br />
year = {2015}, <br />
eprint = {1506.03432},<br />
url = {https://academicworks.cuny.edu/gc_etds/879/}<br />
}<br />
<br />
#CarmodyGitmanHabic2016:Mitchelllike bibtex=@article{CarmodyGitmanHabic2016:Mitchelllike, <br />
author = {Carmody, Erin and Gitman, Victoria and Habič, Miha E.}, <br />
title = {A Mitchell-like order for Ramsey and Ramsey-like cardinals}, <br />
year = {2016}, <br />
eprint = {1609.07645},<br />
}<br />
<br />
#CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal bibtex=@article{CodyGitikHamkinsSchanker2003:TheLeastWeaklyCompactCardinal, <br />
author = {Cody, Brent, Gitik, Moti, Hamkins, Joel David, and Schanker, Jason}, <br />
title = {The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact}, <br />
year = {2013}, <br />
eprint = {1305.5961},<br />
}<br />
<br />
#CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br />
title = "Easton's theorem for Ramsey and strongly Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "9",<br />
pages = "934 - 952",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "10.1016/j.apal.2015.04.006",<br />
url={https://victoriagitman.github.io/files/eastonramsey.pdf},<br />
AUTHOR= {Cody, Brent and Gitman, Victoria},<br />
}<br />
<br />
#Corazza2000:WholenessAxiomAndLaverSequences bibtex=@article{CorazzaAPAL,<br />
author = {Corazza, Paul},<br />
title = {The Wholeness Axiom and Laver sequences},<br />
journal = {Annals of Pure and Applied Logic},<br />
month={October},<br />
year = {2000},<br />
pages={157--260},<br />
}<br />
<br />
#Corazza2003:GapBetweenI3andWA bibtex=@ARTICLE{Corazza2003:GapBetweenI3andWA,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The gap between $\mathrm{I}_3$ and the wholeness axiom},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {179},<br />
YEAR = {2003},<br />
NUMBER = {1},<br />
PAGES = {43--60},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03E55 &#40;03E65)},<br />
MRNUMBER = {MR2028926 &#40;2004k:03100)},<br />
MRREVIEWER = {A. Kanamori},<br />
DOI = {10.4064/fm179-1-4},<br />
URL = {http://dx.doi.org/10.4064/fm179-1-4},<br />
}<br />
<br />
#Corazza2006:TheSpectrumOfElementaryEmbeddings bibtex=@ARTICLE{Corazza2006:TheSpectrumOfElementaryEmbeddings,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The spectrum of elementary embeddings $j : V \to V$},<br />
JOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {139},<br />
MONTH = {May},<br />
YEAR = {2006},<br />
NUMBER = {1--3},<br />
PAGES = {327-399},<br />
DOI = {10.1016/j.apal.2005.06.014},<br />
}<br />
<br />
#Corazza2010:TheAxiomOfInfinityAndJVV bibtex=@ARTICLE{Corazza2010:TheAxiomOfInfinityAndJVV,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The Axiom of Infinity and transformations $j: V \to V$},<br />
JOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {1},<br />
PAGES = {37--84},<br />
DOI = {10.2178/bsl/1264433797},<br />
URL = {https://www.math.ucla.edu/~asl/bsl/1601/1601-002.ps},<br />
}<br />
<br />
#DaghighiPourmahdian2018:PropertiesShelah bibtex=@article{DaghighiPourmahdian2018:PropertiesShelah,<br />
AUTHOR = {Daghighi, Ali Sadegh and Pourmahdian, Massoud},<br />
TITLE = {On Some Properties of Shelah Cardinals},<br />
JOURNAL = {Bull. Iran. Math. Soc.},<br />
FJOURNAL = {Bulletin of the Iranian Mathematical Society},<br />
VOLUME = {44},<br />
YEAR = {2018},<br />
MONTH = {October},<br />
NUMBER = {5},<br />
PAGES = {1117-1124},<br />
DOI = {10.1007/s41980-018-0075-0},<br />
URL = {http://www.alidaghighi.org/wp-content/uploads/2017/08/On-Some-Properties-of-Shelah-Cardinals.pdf}<br />
}<br />
<br />
#Dimonte2017:I0AndRankIntoRankAxioms bibtex=@article {Dimonte2017:I0AndRankIntoRankAxioms,<br />
AUTHOR = {Dimonte, Vincenzo},<br />
TITLE = {I0 and rank-into-rank axioms},<br />
YEAR = {2017},<br />
EPRINT = {1707.02613}<br />
}<br />
<br />
#Dimopoulos2019:WoodinForStrongCompactness bibtex=@article {Dimopoulos2019:WoodinForStrongCompactness,<br />
title={Woodin for strong compactness cardinals},<br />
volume={84},<br />
DOI={10.1017/jsl.2018.67},<br />
number={1},<br />
journal={The Journal of Symbolic Logic},<br />
publisher={Cambridge University Press},<br />
author={Dimopoulos, Stamatis},<br />
year={2019},<br />
pages={301–319},<br />
eprint={1710.05743}<br />
}<br />
<br />
#DoddJensen1982:CoreModel bibtex=@article {MR611394,<br />
AUTHOR = {Dodd, Anthony and Jensen, Ronald},<br />
TITLE = {The core model},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {20},<br />
YEAR = {1981},<br />
NUMBER = {1},<br />
PAGES = {43--75},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03E45 &#40;03C62 03E35)},<br />
MRNUMBER = {611394 &#40;82i:03063)},<br />
MRREVIEWER = {F. R. Drake},<br />
DOI = {10.1016/0003-4843&#40;81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843&#40;81)90011-5},<br />
}<br />
<br />
#DonderKoepke1998:AccessibleJonsson bibtex=@article{DonderKoepke1983:AccessibleJonsson, <br />
author = {Donder, Hans-Dieter and Koepke, Peter},<br />
title = {On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture},<br />
journal = {Annals of Pure and Applied Logic},<br />
year = {1998},<br />
doi = {10.1016/0168-0072&#40;83)90020-9},<br />
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}<br />
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#DonderLevinski1989:PrinciplesRelatedChangsConjecture bibtex=@article{DonderLevinski1989:PrinciplesRelatedChangsConjecture, <br />
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title = {Some principles related to Chang's conjecture},<br />
journal = {Annals of Pure and Applied Logic},<br />
year = {1989},<br />
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doi = {10.1016/0168-0072&#40;89)90030-4},<br />
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AUTHOR = {Drake, Frank},<br />
PUBLISHER = {North-Holland Pub. Co.},<br />
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YEAR = {1974},<br />
SERIES = {Studies in Logic and the Foundations of Mathematics, Volume 76}<br />
}<br />
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#Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals bibtex=@article{Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals,<br />
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journal = {Archive for Mathematical Logic},<br />
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year = {2005},<br />
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}<br />
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#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
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PAGES = {223--226},<br />
ISSN = {0001-5954},<br />
MRCLASS = {04.60},<br />
MRNUMBER = {0141603 &#40;25 \#5001)},<br />
MRREVIEWER = {L. Gillman},<br />
}<br />
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#ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {On the structure of set-mappings},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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}<br />
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#EskrewHayut2016:LocalGlobalChangsConjecture bibtex=@article{EskrewHayut2016:LocalGlobalChangsConjecture,<br />
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year = {2016},<br />
eprint = {1607.04904v4},<br />
}<br />
<br />
#Esser96:GPKAFA bibtex=@article{Esser96:GPKAFA,<br />
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title = {Inconsistency of GPK+AFA},<br />
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volume = {42},<br />
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}<br />
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#Esser96:InterpretationZFCandMKinPositiveTheory bibtex=@article{Esser96:InterpretationZFCandMKinPositiveTheory,<br />
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}<br />
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#Esser99:ConsistencyPositiveTheory bibtex=@article{Esser96:ConsistencyPositiveTheory,<br />
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year = {1999},<br />
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}<br />
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#Esser2000:InconsistencyACwithGPK bibtex=@article{Esser2000:InconsistencyACwithGPK,<br />
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year = {2000},<br />
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url = {http://www.jstor.org/stable/2695086}<br />
}<br />
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#Esser99:ExtensionalityInPositiveTheory bibtex=@article{Esser96:ExtensionalityInPositiveTheory,<br />
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title = {On the axiom of extensionality in the positive set theory},<br />
year = {2003},<br />
journal = {Mathematical Logic Quarterly},<br />
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#EvansHamkins:TransfiniteGameValuesInInfiniteChess bibtex=@ARTICLE{EvansHamkins:TransfiniteGameValuesInInfiniteChess,<br />
AUTHOR = {Evans, C. D. A. and Hamkins, Joel David},<br />
TITLE = {Transfinite game values in infinite chess},<br />
JOURNAL = {},<br />
YEAR = {},<br />
volume = {},<br />
number = {},<br />
pages = {},<br />
month = {},<br />
note = {under review},<br />
eprint = {1302.4377},<br />
url = {http://jdh.hamkins.org/game-values-in-infinite-chess},<br />
abstract = {},<br />
keywords = {},<br />
source = {},<br />
}<br />
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#Feng1990:HierarchyRamsey bibtex=@article{Feng1990:HierarchyRamsey,<br />
title = "A hierarchy of Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "49",<br />
number = "3",<br />
pages = "257 - 277",<br />
year = "1990",<br />
issn = "0168-0072",<br />
doi = "10.1016/0168-0072&#40;90)90028-Z",<br />
author = "Feng, Qi",<br />
}<br />
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#Foreman2010:Handbook bibtex=@book<br />
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editor = {Foreman, Matthew and Kanamori, Akihiro}, <br />
title = {Handbook of Set Theory},<br />
edition = {First}, <br />
publisher = {Springer}, <br />
year = {2010}, <br />
isbn = {978-1-4020-4843-2},<br />
note = {This book is actually a compendium of articles from multiple authors},<br />
url = {http://www.springer.com/mathematics/book/978-1-4020-4843-2},<br />
}<br />
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#FortiHinnion89:ConsitencyProblemPositiveComp bibtex=@article{FortiHinnion89:ConsitencyProblemPositiveComp,<br />
AUTHOR = {Forti, M and Hinnion, R.},<br />
TITLE = {The Consistency Problem for Positive Comprehension Principles},<br />
JOURNAL = {J. Symbolic Logic},<br />
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}<br />
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#Friedman1998:Subtle bibtex=@article{Friedman1998:Subtle,<br />
AUTHOR = {Friedman, Harvey M.},<br />
TITLE = {Subtle cardinals and linear orderings},<br />
YEAR = {1998},<br />
URL = {https://u.osu.edu/friedman.8/files/2014/01/subtlecardinals-1tod0i8.pdf}<br />
}<br />
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#FuchsHamkinsReitz2015:SetTheoreticGeology bibtex=@article{FuchsHamkinsReitz2015:SetTheoreticGeology<br />
title = "Set-theoretic geology",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "4",<br />
pages = "464 - 501",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "https://doi.org/10.1016/j.apal.2014.11.004",<br />
url = "http://www.sciencedirect.com/science/article/pii/S0168007214001225",<br />
author = "Fuchs, Gunter and Hamkins, Joel David and Reitz, Jonas",<br />
title = "Set-theoretic geology",<br />
eprint = "1107.4776",<br />
}<br />
<br />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory &#40;Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 &#40;02H13)},<br />
MRNUMBER = {0376347 &#40;51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo bibtex=@article{GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo,<br />
AUTHOR = {Gitman, Victoria and Hamkins, Joel David},<br />
TITLE = {A model of the generic Vopěnka principle in which the ordinals are not Mahlo},<br />
YEAR = {2018},<br />
EPRINT = {1706.00843v2}<br />
}<br />
<br />
#GitmanJohnstone:IndestructiblyRamsey bibtex=@ARTICLE{GitmanJohnstone:IndestructiblyRamsey,<br />
AUTHOR= {Gitman, Victoria and Johnstone, Thomas A.},<br />
TITLE= {Indestructibility for Ramsey and Ramsey-like cardinals},<br />
NOTE= {In preparation},<br />
URL= {https://victoriagitman.github.io/files/indestructibleramseycardinalsnew.pdf}<br />
}<br />
<br />
#GitmanSchindler:VirtualLargeCardinals bibtex=@ARTICLE{GitmanSchindler:VirtualLargeCardinals,<br />
AUTHOR= {Gitman, Victoria and Shindler, Ralf},<br />
TITLE= {Virtual large cardinals},<br />
URL= {https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf}<br />
}<br />
<br />
#Goldblatt1998: bibtex=@book{Goldblatt1998:ultrafilter,<br />
AUTHOR = {Goldblatt, Robert},<br />
TITLE = {Lectures on the Hyperreals},<br />
PUBLISHER = {Springer},<br />
YEAR = {1998},<br />
}<br />
<br />
#GoldsternShelah1995:BPFA bibtex = @article{GoldsternShelah1995:BPFA,<br />
AUTHOR = {Goldstern, Martin and Shelah, Saharon},<br />
TITLE = {The Bounded Proper Forcing Axiom},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#Golshani2017:EastonLikeInPresenceShelah bibtex=@article{Golshani2017:EastonLikeInPresenceShelah,<br />
AUTHOR = {Golshani, Mohammad},<br />
TITLE = {An Easton like theorem in the presence of Shelah cardinals},<br />
JOURNAL = {M. Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
MONTH = {May},<br />
NUMBER = {3-4},<br />
PAGES = {273-287},<br />
DOI = {10.1007/s00153-017-0528-9},<br />
URL = {http://math.ipm.ac.ir/~golshani/Papers/An%20Easton%20like%20theorem%20in%20the%20presence%20of%20Shelah%20Cardinals.pdf}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article{HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {1771072 &#40;2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 &#40;03E65)},<br />
MRNUMBER = {1816602 &#40;2001m:03102)},<br />
MRREVIEWER = {Ralf-Dieter Schindler},<br />
DOI = {10.1007/s001530050169},<br />
URL = {http://dx.doi.org/10.1007/s001530050169},<br />
eprint = {math/9902079},<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
<!--file = F,--><!--Is it important? It causes an error.--><br />
}<br />
<br />
#Hamkins2008:UnfoldableGCH bibtex=@article{Hamkins2008:UnfoldableGCH, <br />
author = {Hamkins, Joel David},<br />
title = {Unfoldable cardinals and the GCH},<br />
year = {2008},<br />
eprint={math/9909029},<br />
}<br />
<br />
#Hamkins2009:TallCardinals bibtex=@ARTICLE{Hamkins2009:TallCardinals,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Tall cardinals},<br />
JOURNAL = {MLQ Math. Log. Q.},<br />
FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br />
VOLUME = {55},<br />
YEAR = {2009},<br />
NUMBER = {1},<br />
PAGES = {68--86},<br />
ISSN = {0942-5616},<br />
MRCLASS = {03E55 &#40;03E35)},<br />
MRNUMBER = {2489293 &#40;2010g:03083)},<br />
MRREVIEWER = {Carlos A. Di Prisco},<br />
DOI = {10.1002/malq.200710084},<br />
URL = {http://boolesrings.org/hamkins/tallcardinals/},<br />
}<br />
<br />
#HamkinsJohnstone2010:IndestructibleStrongUnfoldability bibtex=@article{HamkinsJohnstone2010:IndestructibleStrongUnfoldability,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Indestructible strong un-foldability},<br />
YEAR = {2010},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
VOLUME = {51},<br />
NUMBER = {3},<br />
PAGES = {291--321}<br />
}<br />
<br />
#HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory bibtex=@article{HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory,<br />
author = {Hamkins, Joel David; Linetsky, David; Reitz, Jonas},<br />
title = {Pointwise Definable Models of Set Theory},<br />
year = {2012},<br />
eprint = {1105.4597}<br />
}<br />
<br />
#HamkinsJohnstone:ResurrectionAxioms bibtex=@article{HamkinsJohnstone:ResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Resurrection axioms and uplifting cardinals},<br />
YEAR = {2014},<br />
url = {http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals/},<br />
eprint = {1307.3602},<br />
}<br />
<br />
#HamkinsJohnstone:BoldfaceResurrectionAxioms bibtex=@article{HamkinsJohnstone:BoldfaceResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Strongly uplifting cardinals and the boldface resurrection axioms},<br />
YEAR = {2014},<br />
eprint = {1403.2788},<br />
}<br />
<br />
#Hamkins2016:TheVopenkaPrincipleIs bibtex=@article{Hamkins2016:TheVopenkaPrincipleIs,<br />
author = {Hamkins, Joel David},<br />
title = {The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme},<br />
year = {2016},<br />
url = {http://jdh.hamkins.org/vopenka-principle-vopenka-scheme/}<br />
eprint = {1606.03778},<br />
}<br />
<br />
#Hauser1991:IndescribableElementaryEmbeddings bibtex=@article{<br />
Hauser1991:IndescribableElementaryEmbeddings,<br />
AUTHOR = {Hauser, Kai},<br />
TITLE = {Indescribable Cardinals and Elementary Embeddings},<br />
VOLUME = {56},<br />
NUMBER = {2},<br />
PAGES = {439 - 457}<br />
YEAR = {1991},<br />
DOI = {10.2307/2274692},<br />
URL = {www.jstor.org/stable/2274692}<br />
}<br />
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#HolySchlicht2017:HierarchyRamseylike bibtex=@article{HolySchlicht2017:HierarchyRamseylike, <br />
author = {Holy, Peter and Schlicht, Philipp}, <br />
title = {A hierarchy of Ramsey-like cardinals}, <br />
year = {2018},<br />
eprint = {1710.10043},<br />
doi = {10.4064/fm396-9-2017},<br />
journal = {Fundamenta Mathematicae},<br />
volume = {242},<br />
pages = {49-74},<br />
url = {https://research-information.bristol.ac.uk/files/185938606/1710.10043.pdf}<br />
}<br />
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#JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson bibtex=@article{<br />
JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson,<br />
AUTHOR = {Jackson, Steve; Ketchersid, Richard; Schlutzenberg, Farmer; Woodin, W. Hugh},<br />
TITLE = {Determinacy and Jónsson cardinals in $L&#40;\mathbb{R})$},<br />
YEAR = {2015},<br />
DOI = {10.1017/jsl.2014.49},<br />
EPRINT = {1304.2323}<br />
}<br />
<br />
#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
SERIES = {Springer Monographs in Mathematics},<br />
NOTE = {The third millennium edition, revised and expanded},<br />
PUBLISHER = {Springer-Verlag},<br />
EDITION = {Third},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<br />
URL = {https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf},<br />
}<br />
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PAGES = {99--275},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
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#Leshem2000:OCDefinableTreeProperty bibtex=@article {Leshem2000:OCDefinableTreeProperty,<br />
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#Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br />
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year = {2010},<br />
doi = {10.1142/S021906131000095X},<br />
URL = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br />
}<br />
<br />
#Woodin2011:SEM2 bibtex=@article{doi:10.1142/S021906131100102X,<br />
author = {Woodin, W. Hugh},<br />
title = {Suitable extender models II: beyond $\omega$-huge},<br />
journal = {Journal of Mathematical Logic},<br />
volume = {11},<br />
number = {02},<br />
pages = {115-436},<br />
year = {2011},<br />
doi = {10.1142/S021906131100102X},<br />
URL = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131100102X}<br />
}<br />
<br />
#Zapletal1996:ANewProofOfKunenInconsistency bibtex=@article {Zapletal1996:ANewProofOfKunenInconsistency,<br />
AUTHOR = {Zapletal, Jindrich},<br />
TITLE = {A new proof of Kunen's inconsistency},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {124},<br />
YEAR = {1996},<br />
NUMBER = {7},<br />
PAGES = {2203--2204},<br />
ISSN = {0002-9939},<br />
CODEN = {PAMYAR},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {MR1317054 &#40;96i:03051)},<br />
MRREVIEWER = {L. Bukovsky},<br />
DOI = {10.1090/S0002-9939-96-03281-9},<br />
URL = {http://dx.doi.org/10.1090/S0002-9939-96-03281-9},<br />
}<br />
</biblio><br />
<br />
== User instructions ==<br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>BartekChomhttp://cantorsattic.info/index.php?title=Reflecting_ordinal&diff=4122Reflecting ordinal2022-05-14T08:39:04Z<p>BartekChom: (compare)</p>
<hr />
<div>[[Category:Lower attic]]<br />
[[Category:Reflection principles]]<br />
: ''Not to be confused with [[reflecting cardinals]].''<br />
''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]].<br />
==Definition==<br />
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite><br />
<br />
([[Indescribable#Indescribable_on_a_set|compare]])<br />
<br />
{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Library&diff=4121Library2022-05-14T08:29:42Z<p>BartekChom: /* Library holdings */ from Reflecting ordinal</p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}[[Category:Cantor's Attic]]<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
Welcome to the library, our central repository for references cited here on Cantor's attic.<br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0460120 &#40;57 \#116)},<br />
MRREVIEWER = {Thomas J. Jech}<br />
}<br />
<br />
#Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod bibtex=@article {Apter2012:SomeApplicationsOfSargsyansEquiconsistencyMethod,<br />
AUTHOR = {Apter, Arthur W.},<br />
TITLE = {Some applications of Sargsyan’s equiconsistency method},<br />
JOURNAL = {Fund. Math.},<br />
VOLUME = {216},<br />
PAGES = {207--222},<br />
}<br />
<br />
#Arai97:P bibtex=@paper{Arai97:P,<br />
title={A sneak preview of proof theory of ordinals},<br />
author={Arai, Toshiyasu},<br />
url={https://www.arxiv.org/abs/1102.0596v1},<br />
year={1997}<br />
}<br />
<br />
#Arai2019:FirstOrderReflection:P bibtex=@paper{Arai2019:FirstOrderReflection,<br />
TITLE = {A simplified ordinal analysis of first-order reflection},<br />
AUTHOR = {Arai, Toshiyasu},<br />
URL = {https://arxiv.org/abs/1907.07611v1},<br />
YEAR = {2019}<br />
}<br />
<br />
#Baaz2011:Kurt bibtex=@book{Baaz2011:Kurt,<br />
title={Kurt Gödel and the Foundations of Mathematics: Horizons of Truth},<br />
author={Baaz, M. and Papadimitriou, C.H. and Putnam, H.W. and Scott, D.S. and Harper, C.L.},<br />
isbn={9781139498432},<br />
url={https://books.google.pl/books?id=Tg0WXU5\_8EgC},<br />
year={2011},<br />
publisher={Cambridge University Press}<br />
}<br />
<br />
#Bagaria2002:AxiomsOfGenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {Axioms of generic absoluteness},<br />
JOURNAL = {Logic Colloquium 2002},<br />
BOOKTITLE = {Logic Colloquium '02: Lecture Notes in Logic 27},<br />
YEAR = {2006},<br />
DOI = {10.1201/9781439865903},<br />
ISBN = {9780429065262},<br />
URL = {https://www.academia.edu/2561575/AXIOMS_OF_GENERIC_ABSOLUTENESS},<br />
}<br />
<br />
#BagariaBosch2004:PFESolovay bibtex=@article {BagariaBosch2004:PFESolovay,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Proper forcing extensions and Solovay models},<br />
JOURNAL = {Archive for Mathematical Logic},<br />
YEAR = {2004},<br />
DOI = {10.1007/s00153-003-0210-2},<br />
URL = {https://www.academia.edu/2561570/Proper_forcing_extensions_and_Solovay_models},<br />
}<br />
<br />
#BagariaBosch2007:GenericAbsoluteness bibtex=@article {Bagaria2002:AxiomsOfGenericAbsoluteness,<br />
AUTHOR = {Bagaria, Joan and Bosch, Roger},<br />
TITLE = {Generic absoluteness under projective forcing},<br />
JOURNAL = {Fundamenta Mathematicae},<br />
YEAR = {2007},<br />
VOLUME = {194},<br />
PAGES = {95-120},<br />
DOI = {10.4064/fm194-2-1},<br />
}<br />
<br />
#Bagaria2012:CnCardinals bibtex=@article{Bagaria2012:CnCardinals,<br />
AUTHOR = {Bagaria, Joan},<br />
TITLE = {$C^{&#40;n)}$-cardinals},<br />
journal = {Archive for Mathematical Logic},<br />
YEAR = {2012},<br />
volume = {51},<br />
number = {3--4},<br />
pages = {213--240},<br />
DOI = {10.1007/s00153-011-0261-8},<br />
URL = {http://www.mittag-leffler.se/sites/default/files/IML-0910f-26.pdf},<br />
eprint = {1908.09664}<br />
}<br />
<br />
#BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@article{BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br />
AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosický, Jiří},<br />
TITLE = {Definable orthogonality classes in accessible categories are small},<br />
journal = {Journal of the European Mathematical Society},<br />
volume = {17},<br />
number = {3},<br />
pages = {549--589},<br />
eprint = {1101.2792}<br />
}<br />
<br />
#BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible bibtex=@article{BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible,<br />
author = {Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi},<br />
title = {Superstrong and other large cardinals are never Laver indestructible},<br />
eprint = {1307.3486},<br />
year = {2013},<br />
journal = {Archive for Mathematical Logic},<br />
volume = {55},<br />
number = {1-2},<br />
pages = {19--35},<br />
url = {http://jdh.hamkins.org/superstrong-never-indestructible/},<br />
doi = {10.1007/s00153-015-0458-3}<br />
}<br />
<br />
#Bagaria2017:LargeCardinalsBeyondChoice bibtex=@article{Bagaria2017:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2017},<br />
url = {https://events.math.unipd.it/aila2017/sites/default/files/BAGARIA.pdf}<br />
}<br />
<br />
#BagariaGitmanSchindler2017:VopenkaPrinciple bibtex=@ARTICLE{BagariaGitmanSchindler2017:VopenkaPrinciple,<br />
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},<br />
TITLE = {Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
NUMBER = {1-2},<br />
PAGES = {1--20},<br />
ISSN = {0933-5846},<br />
MRCLASS = {03E35 &#40;03E55 03E57)},<br />
MRNUMBER = {3598793},<br />
DOI = {10.1007/s00153-016-0511-x},<br />
URL = {https://victoriagitman.github.io/publications/2016/02/10/generic-vopenkas-principle-remarkable-cardinals-and-the-weak-proper-forcing-axiom.html}<br />
}<br />
<br />
#BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice bibtex=@article{BagariaKoellnerWoodin2019:LargeCardinalsBeyondChoice,<br />
author = {Bagaria, Joan and Koellner, Peter and Woodin, W. Hugh},<br />
title = {Large Cardinals beyond Choice},<br />
year = {2019},<br />
journal = {Bulletin of Symbolic Logic},<br />
volume = {25},<br />
number = {3},<br />
pages = {283--318},<br />
url = {https://par.nsf.gov/servlets/purl/10149501},<br />
doi = {10.1017/bsl.2019.28}<br />
}<br />
<br />
#Baumgartner1975:Ineffability bibtex=@incollection{Baumgartner1975:Ineffability,<br />
AUTHOR = {Baumgartner, James},<br />
TITLE = {Ineffability properties of cardinals. I},<br />
BOOKTITLE = {Infinite and finite sets &#40;Colloq., Keszthely, 1973; dedicated to P. Erd&#337;s on his 60th birthday), Vol. I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 &#40;04A20)},<br />
MRNUMBER = {0384553 &#40;52 \#5427)},<br />
MRREVIEWER = {John K. Truss}<br />
}<br />
<br />
#Blass2010:CardinalCharacteristicsHandbook bibtex=@article{Blass2010:CardinalCharacteristicsHandbook,<br />
author = {Blass, Andreas},<br />
title = {Chapter 6: Cardinal characteristics of the continuum},<br />
journal = {Handbook of Set Theory},<br />
editor = {Foreman, Mathew; Kanamori, Akihiro},<br />
year = {2010},<br />
isbn = {1402048432},<br />
publisher = {Springer},<br />
url = {http://www.math.lsa.umich.edu/~ablass/hbk.pdf},<br />
}<br />
<br />
#Blass1976:ExactFunctors bibtex=@article{Blass1976:ExactFunctors,<br />
author = "Blass, Andreas",<br />
fjournal = "Pacific Journal of Mathematics",<br />
journal = "Pacific J. Math.",<br />
number = "2",<br />
pages = "335--346",<br />
publisher = "Pacific Journal of Mathematics, A Non-profit Corporation",<br />
title = "Exact functors and measurable cardinals.",<br />
url = "https://projecteuclid.org:443/euclid.pjm/1102867389",<br />
volume = "63",<br />
year = "1976"<br />
}<br />
<br />
#Boney2017:ModelTheoreticCharacterizations bibtex=@article{BBoney2017:ModelTheoreticCharacterizations,<br />
author = {Boney, Will},<br />
title = {Model Theoretic Characterizations of Large Cardinals},\<br />
year = {2017},<br />
eprint = {1708.07561},<br />
}<br />
<br />
#Bosch2006:SmallDefinablyLargeCardinals bibtex=@article {Bosch2006:SmallDefinablyLargeCardinals,<br />
AUTHOR = {Bosch, Roger},<br />
TITLE = {Small Definably-large Cardinals},<br />
JOURNAL = {Set Theory. Trends in Mathematics},<br />
YEAR = {2006},<br />
PAGES = {55-82},<br />
DOI = {10.1007/3-7643-7692-9_3},<br />
ISBN = {978-3-7643-7692-5},<br />
}<br />
<br />
#Cantor1915:ContributionsFoundingTransfinite bibtex=@book{Cantor1915:ContributionsFoundingTransfinite, <br />
author = {Cantor, Georg}, <br />
title = {Contributions to the Founding of the Theory of Transfinite Numbers}, <br />
editor = {Jourdain, Philip},<br />
note = {Original year was 1915}, <br />
publisher = {Dover}, <br />
address = {New York}, <br />
year = {1955}, <br />
isbn = {978-0-486-60045-1},<br />
url = {http://www.archive.org/details/contributionstot003626mbp},<br />
}<br />
<br />
#Carmody2015:ForceToChangeLargeCardinalStrength bibtex=@article{Carmody2015:ForceToChangeLargeCardinalStrength, <br />
author = {Carmody, Erin Kathryn}, <br />
title = {Force to change large cardinal strength}, <br />
year = {2015}, <br />
eprint = {1506.03432},<br />
url = {https://academicworks.cuny.edu/gc_etds/879/}<br />
}<br />
<br />
#CarmodyGitmanHabic2016:Mitchelllike bibtex=@article{CarmodyGitmanHabic2016:Mitchelllike, <br />
author = {Carmody, Erin and Gitman, Victoria and Habič, Miha E.}, <br />
title = {A Mitchell-like order for Ramsey and Ramsey-like cardinals}, <br />
year = {2016}, <br />
eprint = {1609.07645},<br />
}<br />
<br />
#CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal bibtex=@article{CodyGitikHamkinsSchanker2003:TheLeastWeaklyCompactCardinal, <br />
author = {Cody, Brent, Gitik, Moti, Hamkins, Joel David, and Schanker, Jason}, <br />
title = {The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact}, <br />
year = {2013}, <br />
eprint = {1305.5961},<br />
}<br />
<br />
#CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br />
title = "Easton's theorem for Ramsey and strongly Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "9",<br />
pages = "934 - 952",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "10.1016/j.apal.2015.04.006",<br />
url={https://victoriagitman.github.io/files/eastonramsey.pdf},<br />
AUTHOR= {Cody, Brent and Gitman, Victoria},<br />
}<br />
<br />
#Corazza2000:WholenessAxiomAndLaverSequences bibtex=@article{CorazzaAPAL,<br />
author = {Corazza, Paul},<br />
title = {The Wholeness Axiom and Laver sequences},<br />
journal = {Annals of Pure and Applied Logic},<br />
month={October},<br />
year = {2000},<br />
pages={157--260},<br />
}<br />
<br />
#Corazza2003:GapBetweenI3andWA bibtex=@ARTICLE{Corazza2003:GapBetweenI3andWA,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The gap between $\mathrm{I}_3$ and the wholeness axiom},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {179},<br />
YEAR = {2003},<br />
NUMBER = {1},<br />
PAGES = {43--60},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03E55 &#40;03E65)},<br />
MRNUMBER = {MR2028926 &#40;2004k:03100)},<br />
MRREVIEWER = {A. Kanamori},<br />
DOI = {10.4064/fm179-1-4},<br />
URL = {http://dx.doi.org/10.4064/fm179-1-4},<br />
}<br />
<br />
#Corazza2006:TheSpectrumOfElementaryEmbeddings bibtex=@ARTICLE{Corazza2006:TheSpectrumOfElementaryEmbeddings,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The spectrum of elementary embeddings $j : V \to V$},<br />
JOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {139},<br />
MONTH = {May},<br />
YEAR = {2006},<br />
NUMBER = {1--3},<br />
PAGES = {327-399},<br />
DOI = {10.1016/j.apal.2005.06.014},<br />
}<br />
<br />
#Corazza2010:TheAxiomOfInfinityAndJVV bibtex=@ARTICLE{Corazza2010:TheAxiomOfInfinityAndJVV,<br />
AUTHOR = {Corazza, Paul},<br />
TITLE = {The Axiom of Infinity and transformations $j: V \to V$},<br />
JOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {1},<br />
PAGES = {37--84},<br />
DOI = {10.2178/bsl/1264433797},<br />
URL = {https://www.math.ucla.edu/~asl/bsl/1601/1601-002.ps},<br />
}<br />
<br />
#DaghighiPourmahdian2018:PropertiesShelah bibtex=@article{DaghighiPourmahdian2018:PropertiesShelah,<br />
AUTHOR = {Daghighi, Ali Sadegh and Pourmahdian, Massoud},<br />
TITLE = {On Some Properties of Shelah Cardinals},<br />
JOURNAL = {Bull. Iran. Math. Soc.},<br />
FJOURNAL = {Bulletin of the Iranian Mathematical Society},<br />
VOLUME = {44},<br />
YEAR = {2018},<br />
MONTH = {October},<br />
NUMBER = {5},<br />
PAGES = {1117-1124},<br />
DOI = {10.1007/s41980-018-0075-0},<br />
URL = {http://www.alidaghighi.org/wp-content/uploads/2017/08/On-Some-Properties-of-Shelah-Cardinals.pdf}<br />
}<br />
<br />
#Dimonte2017:I0AndRankIntoRankAxioms bibtex=@article {Dimonte2017:I0AndRankIntoRankAxioms,<br />
AUTHOR = {Dimonte, Vincenzo},<br />
TITLE = {I0 and rank-into-rank axioms},<br />
YEAR = {2017},<br />
EPRINT = {1707.02613}<br />
}<br />
<br />
#Dimopoulos2019:WoodinForStrongCompactness bibtex=@article {Dimopoulos2019:WoodinForStrongCompactness,<br />
title={Woodin for strong compactness cardinals},<br />
volume={84},<br />
DOI={10.1017/jsl.2018.67},<br />
number={1},<br />
journal={The Journal of Symbolic Logic},<br />
publisher={Cambridge University Press},<br />
author={Dimopoulos, Stamatis},<br />
year={2019},<br />
pages={301–319},<br />
eprint={1710.05743}<br />
}<br />
<br />
#DoddJensen1982:CoreModel bibtex=@article {MR611394,<br />
AUTHOR = {Dodd, Anthony and Jensen, Ronald},<br />
TITLE = {The core model},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {20},<br />
YEAR = {1981},<br />
NUMBER = {1},<br />
PAGES = {43--75},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03E45 &#40;03C62 03E35)},<br />
MRNUMBER = {611394 &#40;82i:03063)},<br />
MRREVIEWER = {F. R. Drake},<br />
DOI = {10.1016/0003-4843&#40;81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843&#40;81)90011-5},<br />
}<br />
<br />
#DonderKoepke1998:AccessibleJonsson bibtex=@article{DonderKoepke1983:AccessibleJonsson, <br />
author = {Donder, Hans-Dieter and Koepke, Peter},<br />
title = {On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture},<br />
journal = {Annals of Pure and Applied Logic},<br />
year = {1998},<br />
doi = {10.1016/0168-0072&#40;83)90020-9},<br />
url={https://ac.els-cdn.com/0168007283900209/1-s2.0-0168007283900209-main.pdf?_tid=466bc36a-c95e-11e7-bf33-00000aab0f27&acdnat=1510679420_e0c0ac48663b05db4a42ead08262d38f},<br />
}<br />
<br />
#DonderLevinski1989:PrinciplesRelatedChangsConjecture bibtex=@article{DonderLevinski1989:PrinciplesRelatedChangsConjecture, <br />
author = {Donder, Hans-Dieter and Levinski, Jean-Pierre},<br />
title = {Some principles related to Chang's conjecture},<br />
journal = {Annals of Pure and Applied Logic},<br />
year = {1989},<br />
volume = {45},<br />
pages = {39-101},<br />
doi = {10.1016/0168-0072&#40;89)90030-4},<br />
url={https://ac.els-cdn.com/0168007289900304/1-s2.0-0168007289900304-main.pdf?_tid=2f5a4ffe-e130-11e7-9794-00000aacb361&acdnat=1513298453_24fe48742f365da91523f1174bb74117}<br />
}<br />
<br />
#Drake1974:SetTheory bibtex=@book{Drake1974:SetTheory,<br />
TITLE = {Set Theory: An Introduction to Large Cardinals},<br />
AUTHOR = {Drake, Frank},<br />
PUBLISHER = {North-Holland Pub. Co.},<br />
ISBN = {0444105352, 9780444105356},<br />
YEAR = {1974},<br />
SERIES = {Studies in Logic and the Foundations of Mathematics, Volume 76}<br />
}<br />
<br />
#Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals bibtex=@article{Enayat2005:ModelsOfSetTheoryWithDefinableOrdinals,<br />
author = {Enayat, Ali},<br />
title = {Models of set theory with definable ordinals},<br />
journal = {Archive for Mathematical Logic},<br />
volume = {44},<br />
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}<br />
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#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
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set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
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ISSN = {0001-5954},<br />
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MRNUMBER = {0141603 &#40;25 \#5001)},<br />
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}<br />
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#ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {On the structure of set-mappings},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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}<br />
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title = {On the consistency of local and global versions of Chang's Conjecture},<br />
year = {2016},<br />
eprint = {1607.04904v4},<br />
}<br />
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#Esser96:GPKAFA bibtex=@article{Esser96:GPKAFA,<br />
author = {Esser, Olivier},<br />
title = {Inconsistency of GPK+AFA},<br />
year = {1996},<br />
journal = {Mathematical Logic Quarterly},<br />
doi = {10.1002/malq.19960420109},<br />
volume = {42},<br />
pages = {104--108},<br />
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}<br />
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#Esser96:InterpretationZFCandMKinPositiveTheory bibtex=@article{Esser96:InterpretationZFCandMKinPositiveTheory,<br />
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year = {1997},<br />
journal = {Mathematical Logic Quarterly},<br />
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url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19970430309/abstract}<br />
}<br />
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#Esser99:ConsistencyPositiveTheory bibtex=@article{Esser96:ConsistencyPositiveTheory,<br />
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title = {On the Consistency of a Positive Theory},<br />
year = {1999},<br />
journal = {Mathematical Logic Quarterly},<br />
doi = {10.1002/malq.19990450110},<br />
volume = {45},<br />
pages = {105--116},<br />
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}<br />
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#Esser2000:InconsistencyACwithGPK bibtex=@article{Esser2000:InconsistencyACwithGPK,<br />
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title = {Inconsistency of the Axiom of Choice with the Positive Theory $GPK^+_\infty$},<br />
year = {2000},<br />
month = {Dec.}<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
number = {4},<br />
pages = {1911--1916},<br />
doi = {10.2307/2695086},<br />
url = {http://www.jstor.org/stable/2695086}<br />
}<br />
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#Esser99:ExtensionalityInPositiveTheory bibtex=@article{Esser96:ExtensionalityInPositiveTheory,<br />
author = {Esser, Olivier},<br />
title = {On the axiom of extensionality in the positive set theory},<br />
year = {2003},<br />
journal = {Mathematical Logic Quarterly},<br />
doi = {10.1002/malq.200310009},<br />
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pages = {97--100},<br />
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#EvansHamkins:TransfiniteGameValuesInInfiniteChess bibtex=@ARTICLE{EvansHamkins:TransfiniteGameValuesInInfiniteChess,<br />
AUTHOR = {Evans, C. D. A. and Hamkins, Joel David},<br />
TITLE = {Transfinite game values in infinite chess},<br />
JOURNAL = {},<br />
YEAR = {},<br />
volume = {},<br />
number = {},<br />
pages = {},<br />
month = {},<br />
note = {under review},<br />
eprint = {1302.4377},<br />
url = {http://jdh.hamkins.org/game-values-in-infinite-chess},<br />
abstract = {},<br />
keywords = {},<br />
source = {},<br />
}<br />
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#Feng1990:HierarchyRamsey bibtex=@article{Feng1990:HierarchyRamsey,<br />
title = "A hierarchy of Ramsey cardinals",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "49",<br />
number = "3",<br />
pages = "257 - 277",<br />
year = "1990",<br />
issn = "0168-0072",<br />
doi = "10.1016/0168-0072&#40;90)90028-Z",<br />
author = "Feng, Qi",<br />
}<br />
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#Foreman2010:Handbook bibtex=@book<br />
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author = {Foreman, Matthew and Kanamori, Akihiro},<br />
editor = {Foreman, Matthew and Kanamori, Akihiro}, <br />
title = {Handbook of Set Theory},<br />
edition = {First}, <br />
publisher = {Springer}, <br />
year = {2010}, <br />
isbn = {978-1-4020-4843-2},<br />
note = {This book is actually a compendium of articles from multiple authors},<br />
url = {http://www.springer.com/mathematics/book/978-1-4020-4843-2},<br />
}<br />
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AUTHOR = {Forti, M and Hinnion, R.},<br />
TITLE = {The Consistency Problem for Positive Comprehension Principles},<br />
JOURNAL = {J. Symbolic Logic},<br />
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}<br />
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AUTHOR = {Friedman, Harvey M.},<br />
TITLE = {Subtle cardinals and linear orderings},<br />
YEAR = {1998},<br />
URL = {https://u.osu.edu/friedman.8/files/2014/01/subtlecardinals-1tod0i8.pdf}<br />
}<br />
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#FuchsHamkinsReitz2015:SetTheoreticGeology bibtex=@article{FuchsHamkinsReitz2015:SetTheoreticGeology<br />
title = "Set-theoretic geology",<br />
journal = "Annals of Pure and Applied Logic",<br />
volume = "166",<br />
number = "4",<br />
pages = "464 - 501",<br />
year = "2015",<br />
issn = "0168-0072",<br />
doi = "https://doi.org/10.1016/j.apal.2014.11.004",<br />
url = "http://www.sciencedirect.com/science/article/pii/S0168007214001225",<br />
author = "Fuchs, Gunter and Hamkins, Joel David and Reitz, Jonas",<br />
title = "Set-theoretic geology",<br />
eprint = "1107.4776",<br />
}<br />
<br />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory &#40;Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 &#40;02H13)},<br />
MRNUMBER = {0376347 &#40;51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo bibtex=@article{GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo,<br />
AUTHOR = {Gitman, Victoria and Hamkins, Joel David},<br />
TITLE = {A model of the generic Vopěnka principle in which the ordinals are not Mahlo},<br />
YEAR = {2018},<br />
EPRINT = {1706.00843v2}<br />
}<br />
<br />
#GitmanJohnstone:IndestructiblyRamsey bibtex=@ARTICLE{GitmanJohnstone:IndestructiblyRamsey,<br />
AUTHOR= {Gitman, Victoria and Johnstone, Thomas A.},<br />
TITLE= {Indestructibility for Ramsey and Ramsey-like cardinals},<br />
NOTE= {In preparation},<br />
URL= {https://victoriagitman.github.io/files/indestructibleramseycardinalsnew.pdf}<br />
}<br />
<br />
#GitmanSchindler:VirtualLargeCardinals bibtex=@ARTICLE{GitmanSchindler:VirtualLargeCardinals,<br />
AUTHOR= {Gitman, Victoria and Shindler, Ralf},<br />
TITLE= {Virtual large cardinals},<br />
URL= {https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf}<br />
}<br />
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#Goldblatt1998: bibtex=@book{Goldblatt1998:ultrafilter,<br />
AUTHOR = {Goldblatt, Robert},<br />
TITLE = {Lectures on the Hyperreals},<br />
PUBLISHER = {Springer},<br />
YEAR = {1998},<br />
}<br />
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#GoldsternShelah1995:BPFA bibtex = @article{GoldsternShelah1995:BPFA,<br />
AUTHOR = {Goldstern, Martin and Shelah, Saharon},<br />
TITLE = {The Bounded Proper Forcing Axiom},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
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#Golshani2017:EastonLikeInPresenceShelah bibtex=@article{Golshani2017:EastonLikeInPresenceShelah,<br />
AUTHOR = {Golshani, Mohammad},<br />
TITLE = {An Easton like theorem in the presence of Shelah cardinals},<br />
JOURNAL = {M. Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {56},<br />
YEAR = {2017},<br />
MONTH = {May},<br />
NUMBER = {3-4},<br />
PAGES = {273-287},<br />
DOI = {10.1007/s00153-017-0528-9},<br />
URL = {http://math.ipm.ac.ir/~golshani/Papers/An%20Easton%20like%20theorem%20in%20the%20presence%20of%20Shelah%20Cardinals.pdf}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article{HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {1771072 &#40;2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 &#40;03E65)},<br />
MRNUMBER = {1816602 &#40;2001m:03102)},<br />
MRREVIEWER = {Ralf-Dieter Schindler},<br />
DOI = {10.1007/s001530050169},<br />
URL = {http://dx.doi.org/10.1007/s001530050169},<br />
eprint = {math/9902079},<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 &#40;03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
<!--file = F,--><!--Is it important? It causes an error.--><br />
}<br />
<br />
#Hamkins2008:UnfoldableGCH bibtex=@article{Hamkins2008:UnfoldableGCH, <br />
author = {Hamkins, Joel David},<br />
title = {Unfoldable cardinals and the GCH},<br />
year = {2008},<br />
eprint={math/9909029},<br />
}<br />
<br />
#Hamkins2009:TallCardinals bibtex=@ARTICLE{Hamkins2009:TallCardinals,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Tall cardinals},<br />
JOURNAL = {MLQ Math. Log. Q.},<br />
FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br />
VOLUME = {55},<br />
YEAR = {2009},<br />
NUMBER = {1},<br />
PAGES = {68--86},<br />
ISSN = {0942-5616},<br />
MRCLASS = {03E55 &#40;03E35)},<br />
MRNUMBER = {2489293 &#40;2010g:03083)},<br />
MRREVIEWER = {Carlos A. Di Prisco},<br />
DOI = {10.1002/malq.200710084},<br />
URL = {http://boolesrings.org/hamkins/tallcardinals/},<br />
}<br />
<br />
#HamkinsJohnstone2010:IndestructibleStrongUnfoldability bibtex=@article{HamkinsJohnstone2010:IndestructibleStrongUnfoldability,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Indestructible strong un-foldability},<br />
YEAR = {2010},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
VOLUME = {51},<br />
NUMBER = {3},<br />
PAGES = {291--321}<br />
}<br />
<br />
#HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory bibtex=@article{HamkinsLinetskyReitz2012:PointwiseDefinableModelsOfSetTheory,<br />
author = {Hamkins, Joel David; Linetsky, David; Reitz, Jonas},<br />
title = {Pointwise Definable Models of Set Theory},<br />
year = {2012},<br />
eprint = {1105.4597}<br />
}<br />
<br />
#HamkinsJohnstone:ResurrectionAxioms bibtex=@article{HamkinsJohnstone:ResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Resurrection axioms and uplifting cardinals},<br />
YEAR = {2014},<br />
url = {http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals/},<br />
eprint = {1307.3602},<br />
}<br />
<br />
#HamkinsJohnstone:BoldfaceResurrectionAxioms bibtex=@article{HamkinsJohnstone:BoldfaceResurrectionAxioms,<br />
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},<br />
TITLE = {Strongly uplifting cardinals and the boldface resurrection axioms},<br />
YEAR = {2014},<br />
eprint = {1403.2788},<br />
}<br />
<br />
#Hamkins2016:TheVopenkaPrincipleIs bibtex=@article{Hamkins2016:TheVopenkaPrincipleIs,<br />
author = {Hamkins, Joel David},<br />
title = {The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme},<br />
year = {2016},<br />
url = {http://jdh.hamkins.org/vopenka-principle-vopenka-scheme/}<br />
eprint = {1606.03778},<br />
}<br />
<br />
#Hauser1991:IndescribableElementaryEmbeddings bibtex=@article{<br />
Hauser1991:IndescribableElementaryEmbeddings,<br />
AUTHOR = {Hauser, Kai},<br />
TITLE = {Indescribable Cardinals and Elementary Embeddings},<br />
VOLUME = {56},<br />
NUMBER = {2},<br />
PAGES = {439 - 457}<br />
YEAR = {1991},<br />
DOI = {10.2307/2274692},<br />
URL = {www.jstor.org/stable/2274692}<br />
}<br />
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#HolySchlicht2017:HierarchyRamseylike bibtex=@article{HolySchlicht2017:HierarchyRamseylike, <br />
author = {Holy, Peter and Schlicht, Philipp}, <br />
title = {A hierarchy of Ramsey-like cardinals}, <br />
year = {2018},<br />
eprint = {1710.10043},<br />
doi = {10.4064/fm396-9-2017},<br />
journal = {Fundamenta Mathematicae},<br />
volume = {242},<br />
pages = {49-74},<br />
url = {https://research-information.bristol.ac.uk/files/185938606/1710.10043.pdf}<br />
}<br />
<br />
#JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson bibtex=@article{<br />
JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson,<br />
AUTHOR = {Jackson, Steve; Ketchersid, Richard; Schlutzenberg, Farmer; Woodin, W. Hugh},<br />
TITLE = {Determinacy and Jónsson cardinals in $L&#40;\mathbb{R})$},<br />
YEAR = {2015},<br />
DOI = {10.1017/jsl.2014.49},<br />
EPRINT = {1304.2323}<br />
}<br />
<br />
#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
SERIES = {Springer Monographs in Mathematics},<br />
NOTE = {The third millennium edition, revised and expanded},<br />
PUBLISHER = {Springer-Verlag},<br />
EDITION = {Third},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<br />
URL = {https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf},<br />
}<br />
<br />
#JensenKunen1969:Ineffable bibtex=@unpublished{JensenKunen1969:Ineffable,<br />
AUTHOR={Jensen, Ronald and Kunen, Kenneth},<br />
TITLE={Some combinatorial properties of $L$ and $V$},<br />
YEAR={1969},<br />
URL={http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html},<br />
}<br />
<br />
#Kanamori1977:EvolutionLargeCardinals bibtex=@incollection {#Kanamori1977:EvolutionLargeCardinals,<br />
AUTHOR = {Kanamori, Akihiro and Magidor, Menachem},<br />
TITLE = {The evolution of large cardinal axioms in set theory},<br />
BOOKTITLE = {Higher set theory &#40;Proc. Conf., Math. Forschungsinst.,<br />
Oberwolfach, 1977)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {669},<br />
PAGES = {99--275},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1978},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {520190 &#40;80b:03083)},<br />
MRREVIEWER = {J. L. Bell},<br />
url = {http://math.bu.edu/people/aki/e.pdf},<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''&#40;1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},<br />
}<br />
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#KanamoriAwerbuchFriedlander1990:Compleat0Dagger bibtex=@article{KanamoriAwerbuchFriedlander1990:Compleat0Dagger,<br />
author = {Kanamori, Akihiro and Awerbuch-Friedlander, Tamara},<br />
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#Laver1997:Implications bibtex=@article {Laver1997:Implications,<br />
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#Leshem2000:OCDefinableTreeProperty bibtex=@article {Leshem2000:OCDefinableTreeProperty,<br />
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#Maddy88:BelAxiomsI bibtex=@article{Maddy88:BelAxiomsI,<br />
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#Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br />
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#Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br />
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#NielsenWelch2018:GamesRamseylike bibtex=@article{NielsenWelch2018:GamesRamseylike, <br />
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}<br />
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#Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br />
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}<br />
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</biblio><br />
<br />
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Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>BartekChomhttp://cantorsattic.info/index.php?title=Mahlo&diff=4120Mahlo2022-05-13T18:25:38Z<p>BartekChom: Mahlo on a set</p>
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<div>{{DISPLAYTITLE: Mahlo cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
A cardinal $\kappa$ is ''Mahlo'' if and only if it is [[inaccessible]] and the [[regular]] cardinals below $\kappa$ form a [[stationary]] subset of $\kappa$. Equivalently, $\kappa$ is Mahlo if it is [[regular]] and the [[inaccessible]] cardinals below $\kappa$ are stationary. <br />
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* Every Mahlo cardinal $\kappa$ is inaccessible, and indeed hyper-inaccessible and hyper-hyper-inaccessible, up to degree $\kappa$, and a limit of such cardinals. <br />
* If $\kappa$ is Mahlo, then it is Mahlo in any inner model, since the concept of stationarity is similarly downward absolute.<br />
* A cardinal is ''greatly inaccessible'' iff there is a uniform, normal [[filter]] on it, closed under the inaccessible limit point operator $\mathcal{I}&#40;X) = \{α ∈ X : α$ is an inaccessible limit point of $X\}$. This property is equivalent to being Mahlo and analogous to being [[Mahlo#greatly Mahlo|greatly Mahlo]].<cite>Carmody2015:ForceToChangeLargeCardinalStrength</cite><br />
* A cardinal $\kappa$ is Mahlo iff there exists a nontrivial $\kappa$-complete $\kappa$-normal ideal on it. [https://theory.stanford.edu/~tingz/talks/mahlo.pdf]<br />
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[[Ord is Mahlo]] is a scheme asserting that regular cardinals form a stationary class. It is weaker than the existence of a Mahlo cardinal &#40;or even pseudo $0$-[[uplifting]] cardinal<cite>HamkinsJohnstone:ResurrectionAxioms</cite>).<br />
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Mahlo cardinals belong to the oldest large cardinals together with inaccessible and measurable. ''Please add more history.''<br />
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==Weakly Mahlo==<br />
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A cardinal $\kappa$ is ''weakly Mahlo'' if it is [[regular]] and the set of [[regular]] cardinals below $\kappa$ is [[stationary]] in $\kappa$. If $\kappa$ is a [[strong limit]] and hence also [[inaccessible]], this is equivalent to $\kappa$ being Mahlo, since the [[strong limit]] cardinals form a closed unbounded subset in any [[inaccessible]] cardinal. In particular, under the [[generalized continuum hypothesis | GCH]], a cardinal is weakly Mahlo if and only if it is Mahlo. But in general, the concepts can differ, since adding an enormous number of Cohen reals will preserve all weakly Mahlo cardinals, but can easily destroy strong limit cardinals. Thus, every Mahlo cardinal can be made weakly Mahlo but not Mahlo in a forcing extension in which the continuum is very large. Nevertheless, every weakly Mahlo cardinal is Mahlo in any inner model of the GCH.<br />
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Equivalently, $\kappa$ is weakly Mahlo iff for all functions $f:\kappa\rightarrow\kappa$, there exists an uncountable regular $\kappa'\in\kappa$ such that $\kappa'$ is closed under $f$.<!--http://www1.maths.leeds.ac.uk/~rathjen/Ord_Notation_Weakly_Mahlo.pdf#page=2--><br />
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==Hyper-Mahlo etc.==<br />
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A cardinal $\kappa$ is ''$1$-Mahlo'' if the set of Mahlo cardinals is stationary in $\kappa$. This is a strictly stronger notion than merely asserting that $\kappa$ is a Mahlo limit of Mahlo cardinals, since in fact every $1$-Mahlo cardinal is a limit of such Mahlo-limits-of-Mahlo cardinals. &#40;So there is an entire hierarchy of limits-of-limits-of-Mahloness between the Mahlo cardinals and the $1$-Mahlo cardinals.) More generally, $\kappa$ is $\alpha$-Mahlo if it is Mahlo and for each $\beta\lt\alpha$ the class of $\beta$-Mahlo cardinals is stationary in $\kappa$. The cardinal $\kappa$ is ''hyper-Mahlo'' if it is $\kappa$-Mahlo. One may proceed to define the concepts of $\alpha$-hyper${}^\beta$-Mahlo by iterating this concept, iterating the stationary limit concept. All such levels are swamped by the [[weakly compact]] cardinals, which exhibit all the desired degrees of hyper-Mahloness and more:<br />
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Meta-ordinal terms are terms like $Ω^α · β + Ω^γ · δ +· · ·+Ω^\epsilon · \zeta + \theta$ where $α, β...$ are ordinals. They are ordered as if $Ω$ were an ordinal greater then all the others, this ordering being a concept that can be formalized using term what Williams calls "arithmetic term symbols"<!--Source?-->, which are ordered tuples of ordinals. $&#40;Ω · α + β)$-Mahlo denotes $β$-hyper${}^α$-Mahlo, $Ω^2$-Mahlo denotes hyper${}^\kappa$-Mahlo $\kappa$ etc. Every weakly compact cardinal $\kappa$ is $\Omega^α$-Mahlo for all $α<\kappa$ and probably more. Similar hierarchy exists for [[inaccessible]] cardinals below Mahlo. All such properties can be killed softly by forcing to make them any weaker properties from this family.<cite>Carmody2015:ForceToChangeLargeCardinalStrength</cite><br />
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==$\Sigma_n$-Mahlo etc.==<br />
A regular cardinal $κ$ is $Σ_n$-Mahlo &#40;$Π_n$-Mahlo, respectively) if every club in $κ$ that is $Σ_n$-definable &#40;$Π_n$-definable, respectively) in $H&#40;κ)$ contains an inaccessible cardinal. A regular cardinal $κ$ is $Σ_ω$-Mahlo if every club subset of $κ$ that is definable &#40;with parameters) in $H&#40;κ)$ contains an inaccessible cardinal.<br />
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Every $Π_1$-Mahlo cardinal is an inaccessible limit of inaccessible cardinals. For Mahlo $κ$, the set of $Σ_ω$-Mahlo cardinals is stationary on $κ$.<br />
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In <cite>Bosch2006:SmallDefinablyLargeCardinals</cite> it is shown that every $Σ_ω$-[[weakly compact]] cardinal is $Σ_ω$-Mahlo and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.<br />
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These properties are connected with [[axioms of generic absoluteness]]. For example:<br />
* There is a, provably in ZFC, ccc poset $\mathbb{P}$ which is both $Σ_2$ and $Π_2$ definable in $H&#40;ω_1&#x29;$, without parameters, and for which the axiom $\mathcal{A}&#40;H&#40;ω_1&#x29;, Σ_3, \mathbb{P}&#x29;$ fails if $ω_1$ is not a $Π_1$-Mahlo cardinal in $L$.<br />
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with both $\mathcal{A}&#40;H&#40;ω_1&#x29;, \underset{\sim}{Σ_ω}, Γ&#x29;$ and $\mathcal{A}&#40;L&#40;\mathbb{R}&#x29;, \underset{\sim}{Σ_ω}, Γ&#x29;$ where $Γ$ is the class of absolutely-ccc projective posets.<br />
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* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of both $\mathcal{A}&#40;H&#40;ω_1&#x29;, \underset{\sim}{Σ_ω}, Γ&#x29;$ and $\mathcal{A}&#40;L&#40;\mathbb{R}&#x29;, \underset{\sim}{Σ_ω}, Γ&#x29;$ where $Γ$ is the class of strongly proper projective posets.<br />
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This section from<cite>BagariaBosch2004:PFESolovay</cite><cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite><br />
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==Mahlo on a set==<br />
The ordinal $\alpha$ is called Mahlo on $X\subseteq\mathrm{Ord}$ iff for every $f:\alpha\rightarrow\alpha$ there is a $\beta > 0$ closed under $f$ such that $\beta \in X\cap\alpha$.<br />
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Mahloness on $X$ is equivalent to $\Pi_2^0$-[[indescribable|indescribablity]] on $X$ and to $\Pi_0^1$-indescribablity on $X$.&#40;theorem 1.3 ii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
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{{References}}</div>BartekChomhttp://cantorsattic.info/index.php?title=Indescribable&diff=4119Indescribable2022-05-13T18:11:28Z<p>BartekChom: Indescribable on a set</p>
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<div>{{DISPLAYTITLE:Indescribable cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Reflection principles]]<br />
[[File:IndescribableStructure.png | thumb | The Structure of Indescribability in Consistency Strength]]<br />
A cardinal $\kappa$ is '''indescribable''' if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order sentence $\phi$:<br />
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$$\forall S\subseteq V_{\kappa}(\langle V_{\kappa+n};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+n};\in,S\cap V_{\alpha}\rangle\models\phi))$$<br />
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Likewise for $\Sigma_{m}^n$-indescribable cardinals.<br />
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Here are some other equivalent definitions:<br />
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*A cardinal $\kappa$ is $\Pi_m^n$-indescribable for $n>0$ iff for every $\Pi_m$ first-order unary formula $\phi$:<br />
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$$\forall S\subseteq V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha<\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))$$<br />
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*A cardinal $\kappa$ is $\Pi_m^n$-indescribable iff for every $\Pi_m$ $n+1$-th-order sentence $\phi$:<br />
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$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
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In other words, if a cardinal is $\Pi_{m}^n$-indescribable, then every $n+1$-th order logic statement that is $\Pi_m$ expresses the reflection of $V_{\kappa}$ onto $V_{\alpha}$. This exercises the fact that these cardinals are so large they almost resemble the order of $V$ itself. This definition is similar to that of [[shrewd]] cardinals, an extension of indescribable cardinals.<br />
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==Variants==<br />
===Language===<br />
'''$Q$-indescribable''' cardinals are those which have the property that for every $Q$-sentence $\phi$:<br />
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$$\forall S\subseteq V_\kappa(\langle V_\kappa;\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_\alpha;\in,S\cap V_\alpha\rangle\models\phi))$$<br />
By extending languages of set theory, we can obtain many various sets $Q$, defining many different varieties of indescribability.<br />
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One example is Bagaria's extension of the Levy hierarchy using infinitary logic. This allows us to speak about $\Pi^1_\xi$-formulae for ordinal $\xi$, so using it we can speak about $\Pi^1_\xi$-indescribability. [https://arxiv.org/pdf/1907.13540.pdf#page=12]<br />
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===Higher-order===<br />
'''Totally indescribable''' cardinals are $\Pi_m^n$-indescribable for every natural $m$ and $n$ (equivalently $\Sigma_m^n$-indescribable for every natural m and n, equivalently $\Delta_m^n$-indescribable for every natural $m$ and $n$). This means that every (finitary) formula made from quantifiers, $\in$ and a subset of $V_{\kappa}$ reflects from $V_{\kappa}$ onto a smaller rank. <br />
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'''$\beta$-indescribable''' cardinals are those which have the property that for every first order sentence $\phi$:<br />
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$$\forall S\subseteq V_\kappa(\langle V_{\kappa+\beta};\in,S\rangle\models\phi\rightarrow\exists\alpha<\kappa(\langle V_{\alpha+\beta};\in,S\cap V_\alpha\rangle\models\phi))$$<br />
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There is no $\kappa$ which is $\kappa$-indescribable. A cardinal is $\Pi_{<\omega}^m$-indescribable iff it is $m$-indescribable for finite $m$. Every $\omega$-indescribable cardinal is totally indescribable.<br />
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===Indescribable on a set===<br />
&#40;from <cite>RichterAczel1974:InductiveDefinitions</cite>)<br />
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Language $\mathcal{L}$ has, among others, variables and quantifiers for all finite types &#40;where variables of type 0 range over individuals, of type 1 – over sets of individuals etc.). &#40;§1)<br />
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We say that $\alpha\in\mathrm{Ord}$ reflects a sentence $\varphi$ of $\mathcal{L}$ on $X\subseteq\mathrm{Ord}$ iff $\alpha \models \varphi \implies \exists_{\beta \in X \cap \alpha} \beta \models \phi$. (definition 1.1<!--first part-->)<br />
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We call $\alpha$ $Q$-indescribable on $X$ iff $\alpha$ reflects on $X$ every $Q$-sentence of $\mathcal{L}$. (definition 1.1<!--second part-->)<br />
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Reflection/indescribability on $\mathrm{Ord}$ is simply called reflection/indescribability.<br />
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==Facts==<br />
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Here are some known facts about indescribability:<br />
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<!--$\Pi_2^0$-indescribability is equivalent to being [[uncountable]] and [[regular]]. &#40;theorem 1.2)<cite>RichterAczel1974:InductiveDefinitions</cite> ...... a nonequivalent definition? ......--><br />
$\Pi_2^0$-indescribability is equivalent to [[inaccessible|strong inaccessibility]], $\Sigma_1^1$-indescribablity, $\Pi_n^0$-indescribability given any $n>1$, and $\Pi_0^1$-indescribability.<cite>Kanamori2009:HigherInfinite</cite> $\Pi_1^1$-indescribability is equivalent to [[weakly compact|weak compactness]]. <cite>Jech2003:SetTheory</cite>,<cite>Kanamori2009:HigherInfinite</cite><br />
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The property of being a limit &#40;$\alpha = \sup &#40;X \cap \alpha)$) is equivalent to $\Pi_0^0$-indescribablity on $X$ and to $\Sigma_2^0$-indescribablity on $X$. [[Mahlo]]ness on $X$ is equivalent to $\Pi_2^0$-indescribablity on $X$ and to $\Pi_0^1$-indescribablity on $X$. $\Pi_n^1$-indescribablity on $X$ is equivalent to $\Sigma_{n+1}^1$-indescribablity on $X$. &#40;theorem 1.3 i-iii)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
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If $m>2$ or $n>0$, $\Pi_m^n$-indescribablity on $X$ is equivalent to $\Pi_m^n$-indescribablity on $X\cap\mathrm{Rg}$. If $m>3$ or $n>0$, $\Sigma_m^n$-indescribablity on $X$ is equivalent to $\Sigma_m^n$-indescribablity on $X\cap\mathrm{Rg}$. &#40;$\mathrm{Rg}$ is the class of [[regular]] cardinals.) &#40;theorem 1.3 iv)<cite>RichterAczel1974:InductiveDefinitions</cite><br />
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$\Pi_n^m$-indescribablity is equivalent to $m$-$\Pi_n$-shrewdness (similarly with $\Sigma_n^m$). <cite>Rathjen2006:OrdinalAnalysis</cite><br />
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[[Ineffable]] cardinals are $\Pi^1_2$-indescribable and limits of totally indescribable cardinals. <cite>JensenKunen1969:Ineffable</cite><br />
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$\Pi_n^1$-indescribability is equivalent to $\Sigma_{n+1}^1$-Indescribability. <cite>Kanamori2009:HigherInfinite</cite><br />
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If $m>1$, $\Pi_{n+1}^m$-indescribability is stronger (consistency-wise) than $\Sigma_n^m$ and $\Pi_n^m$-indescribability; every $\Pi_{n+1}^m$-indescribable cardinal is also both $\Sigma_n^m$ and $\Pi_n^m$-indescribable and a stationary limit of such for $m>1$.<cite>Kanamori2009:HigherInfinite</cite> If $m>1$, the least $\Pi_n^m$-indescribable cardinal is less than the least $\Sigma_n^m$-indescribable cardinal, which is in turn less than the least $\Pi_{n+1}^m$-indescribable cardinal.<cite>Kanamori2009:HigherInfinite</cite><br />
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If $\kappa$ is $Π_n$-[[Ramsey]], then $\kappa$ is $Π_{n+1}^1$-indescribable. If $X\subseteq\kappa$ is a $Π_n$-Ramsey subset, then $X$ is in the $Π_{n+1}^1$-indescribable filter.<cite>Feng1990:HierarchyRamsey</cite> If $\kappa$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.<cite>HolySchlicht2017:HierarchyRamseylike</cite><br />
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Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2n+2}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite> Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.<cite>NielsenWelch2018:GamesRamseylike</cite> Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.<cite>NielsenWelch2018:GamesRamseylike</cite><br />
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Every [[measurable]] cardinal is $\Pi_1^2$-indescribable. Although, the least measurable is $\Sigma_1^2$-describable. <cite>Jech2003:SetTheory</cite><br />
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Every critical point of a nontrivial elementary embedding $j:M\rightarrow M$ for some transitive inner model $M$ of [[ZFC]] is totally indescribable in $M$. (For example, [[rank-into-rank]] cardinals, [[Zero sharp|$0^{\#}$]] cardinals, and [[Zero dagger|$0^{\dagger}$]] cardinals). <cite>Jech2003:SetTheory</cite><br />
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If $2^\kappa\neq\kappa^+$ for some $\Pi_1^2$-indescribable cardinal, then there is a smaller $\lambda$ such that $2^\lambda\neq\lambda^+$. However, assuming the consistency of the existence of a $\Pi_n^1$-indescribable cardinal $\kappa$, it is consistent for $\kappa$ to be the least cardinal such that $2^\kappa\neq\kappa^+$. <cite>Hauser1991:IndescribableElementaryEmbeddings</cite><br />
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Transfinite $Π^1_α$-indescribable has been defined via finite games and it turns out that for infinite $α$, if $κ$ is $Π_α$-[[Ramsey]], then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$.<cite>SharpeWelch2011:GreatlyErdosChang</cite><br />
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$\mathrm{ZFC} + \mathrm{BTEE}$ ([[Basic Theory of Elementary Embeddings]]) proves that the critical point of $j$ is totally indescribable.<cite>Corazza2006:TheSpectrumOfElementaryEmbeddings</cite><br />
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{{References}}</div>BartekChom