http://cantorsattic.info/api.php?action=feedcontributions&user=Zetapology&feedformat=atom Cantor's Attic - User contributions [en] 2019-08-21T23:22:13Z User contributions MediaWiki 1.24.4 http://cantorsattic.info/index.php?title=Vopenka&diff=2953 Vopenka 2019-06-10T19:17:55Z <p>Zetapology: /* Vopěnka cardinals */ making it true</p> <hr /> <div>{{DISPLAYTITLE: Vopěnka's principle and Vopěnka cardinals}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Reflection principles]]<br /> Vopěnka's principle is a large cardinal axiom at the upper end of the large cardinal hierarchy that is particularly notable for its applications to category theory. <br /> In a set theoretic setting, the most common definition is the following:<br /> &lt;blockquote&gt;<br /> For any language $\mathcal{L}$ and any proper class $C$ of $\mathcal{L}$-structures, there are distinct structures $M, N\in C$ and an [[elementary embedding]] $j:M\to N$.<br /> &lt;/blockquote&gt;<br /> For example, taking $\mathcal{L}$ to be the language with one unary and one binary predicate, we can consider for any ordinal $\eta$ the class of structures<br /> $\langle V_{\alpha+\eta},\{\alpha\},\in\rangle$, and conclude from Vopěnka's principle that a cardinal that is at least $\eta$-[[extendible]] exists.<br /> In fact if Vopěnka's principle holds then there are a proper class of extendible cardinals; bounding the strength of the axiom from above, we have that<br /> if $\kappa$ is [[huge#Almost huge|almost huge]], or even [[high-jump|almost-high-jump]], then $V_\kappa$ satisfies Vopěnka's principle. <br /> <br /> ==Formalizations==<br /> <br /> As stated above and from the point of view of ZFC, this is actually an axiom schema, as we quantify over proper classes, which from a purely ZFC perspective means definable proper classes. A somewhat stronger alternative is to view Vopěnka's principle as an axiom in second-order set theory capable to dealing with proper classes, such as von Neumann-Gödel-Bernays set theory. This is a strictly stronger assertion. [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably] Finally, one may relativize the principle to a particular cardinal, leading to the concept of a Vopěnka cardinal. <br /> <br /> == Vopěnka cardinals == <br /> <br /> An inaccessible cardinal $\kappa$ is a ''Vopěnka cardinal'' if and only if $V_\kappa$ satisfies Vopěnka's principle, that is, where we interpret the proper classes of $V_\kappa$ as the subsets of $V_\kappa$ of cardinality $\kappa$. Because of a characterization of Vopěnka's principle in terms of graphs, a cardinal $\kappa$ is Vopěnka if and only if $\kappa$ is inaccessible and any set $\kappa$-sized set $G$ of $&lt;\kappa$-sized nonisomorphic graphs has some $g_0$ and $g_1$ with $g_0$ a proper subgraph of $g_1$. (Need to cite sources)<br /> <br /> Perlmutter &lt;cite&gt;Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge&lt;/cite&gt; proved that a cardinal is a Vopěnka cardinal if and only if it is a [[Woodin|Woodin for supercompactness]] cardinal. <br /> <br /> As we mentioned above, every almost huge cardinal is a Vopěnka cardinal.<br /> <br /> ==Equivalent statements==<br /> <br /> === $C^{(n)}$-extendible cardinals ===<br /> <br /> The schema form of Vopěnka's principle is equivalent to the existence of a proper class of $C^{(n)}$-[[extendible]] cardinals for every $n$; indeed there is a level-by-level stratification of Vopěnka's principle, with Vopěnka's principle for a $\Sigma_{n+2}$-definable class corresponds to the existence of a $C^{(n)}$-extendible cardinal greater than the ranks of the parameters.<br /> &lt;CITE&gt;BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses&lt;/CITE&gt;<br /> <br /> === Strong Compactness of Logics ===<br /> <br /> Vopěnka's principle is equivalent to the following statement about logics as well:<br /> <br /> For every logic $\mathcal{L}$, there is a cardinal $\mu_{\mathcal{L}}$ such that for any language $\tau$ and any $\mathcal{L}(\tau)$-theory $T$, $T$ is satisfiable if and only if every $t\subseteq T$ such that $|t|&lt;\mu_{\mathcal{L}}$ is satisfiable. &lt;cite&gt;Makowsky1985:CompactLogics&lt;/cite&gt;<br /> <br /> This $\mu_{\mathcal{L}}$ is called the strong compactness cardinal of $\mathcal{L}$. Vopěnka's principle therefore is equivalent to every logic having a strong compactness cardinal. This is very similar in definition to the Löwenheim–Skolem number of $\mathcal{L}$, although it is not guaranteed to exist. <br /> <br /> Here are some examples of strong compactness cardinals of specific logics:<br /> <br /> *If $\kappa\leq\lambda$ and $\lambda$ is [[strongly compact]] or $\aleph_0$, then the strong compactness cardinal of [[infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] is at most $\lambda$.<br /> *Similarly, if $\kappa\leq\lambda$ and $\lambda$ is [[extendible]], then for any natural number $n$, the strong compactness cardinal of $\mathcal{L}^n_{\kappa,\kappa}$ ($\mathcal{L}_{\kappa,\kappa}$ with $n+1$-th order logic) is at most $\lambda$. Therefore for any natural number $n$, the strong compactness cardinal of $n+1$-th order finitary logic is at most the least extendible cardinal.<br /> <br /> === Locally Presentable Categories ===<br /> <br /> Vopěnka's principle is equivalent to the axiom stating &quot;no large full subcategory $C$ of any locally presentable category is discrete.&quot; (Sources needed). Equivalently, no large full subcategory of Graph (the category of all graphs) is discrete; that is, for any proper class of simple directed graphs, there is at least one pair of nonequal graphs $G$ and $H$ in the class such that $G$ is a subgraph of $H$. This is a $\Pi^1_1$ statement, so the least Vopěnka cardinals are not even [[weakly compact]] (although the least weakly compact cardinal is much, much, much smaller than the least Vopěnka cardinal, if it exists).<br /> <br /> Intuitively, a &quot;category&quot; is just a class of mathematical objects with some notion of &quot;morphism&quot;, &quot;homomorphism&quot;, &quot;isomorphism&quot;, (etc.). For example, in Set, the category of all sets, homomorphisms are just injections, and isomorphisms are bijections. In categories of groups and models, homomorphisms and isomorphisms share their actual names.<br /> <br /> A &quot;locally small category&quot; $C$ is one with only set-many morphisms between any two objects of $C$. This is one where the objects of $C$ behave &quot;set-like&quot; in the sense that, usually, the number of morphisms between two set-sized objects is at most the number of functions between their universes (like in groups and in graphs). A &quot;locally presentable category&quot; is a locally small category with a couple more really nice properties; you can &quot;generate&quot; all of the objects from set-many objects in the category.<br /> <br /> Vopěnka's principle intuitively states that if you have a locally presentable category $C$, then any proper class of objects of $C$ has some nonisomorphic objects $c$ and $d$ where $c$ has a morphism into $d$.<br /> <br /> === Woodin cardinals ===<br /> <br /> There is a strange connection between the [[Woodin]] cardinals and the Vopěnka cardinals. In particular, Vopěnkaness is equivalent to two strengthening variants of Woodinness, namely the [[Woodin#Shelah|Woodin for Supercompactness]] cardinals and the [[n-fold Woodin|$2$-fold Woodin]] cardinals. As a result, every Vopěnka cardinal is Woodin.<br /> <br /> === Elementary Embeddings Between Ranks ===<br /> <br /> An equivalent statement to Vopěnka's principle is that for any proper class $C\subseteq ORD$, there are $\alpha\in C$, $\beta\in C$, and a nontrivial [[elementary embedding]] $j:\langle V_\alpha;\in,P\rangle\rightarrow\langle V_\beta;\in,P\rangle$. Vopěnka's principle quite obviously implies this. The reason the converse holds is because every elementary embedding can be &quot;encoded&quot; (in a sense) into one of these. For more information, see &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;.<br /> <br /> ==Other points to note==<br /> <br /> Whilst Vopěnka cardinals are very strong in terms of consistency strength, a Vopěnka cardinal need not even be [[weakly compact]]. Indeed, the definition of a Vopěnka cardinal is a $\Pi^1_1$ statement over $V_\kappa$ (Vopěnka's principle itself is $\Pi^1_1$), and [[indescribable|$\Pi^1_1$-indescribability]] is one of the equivalent definitions of weak compactness. Thus, the least weakly compact Vopěnka cardinal must have (many) other Vopěnka cardinals less than it.<br /> <br /> ==External links==<br /> <br /> * [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably Math Overflow question and answer about formalisations]<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Talk:Huge&diff=2941 Talk:Huge 2019-05-10T06:06:34Z <p>Zetapology: added rebuttal to amateur analysis</p> <hr /> <div>== $\omega$-almost hugeness underexplored? ==<br /> <br /> The page briefly mentions the $\omega$-almost huge cardinals, but then immediately moves on to $\omega$-huge cardinals without mentioning anything about the properties or possibility/impossibility of $\omega$-almost huge cardinals. Furthermore, none of the other pages in this wiki make any mention of $\omega$-almost hugeness. Is it consistent for them to exist, how strong are they, and is it known how they relate to other large cardinals? [[User:Eaglgenes101|Eaglgenes101]] ([[User talk:Eaglgenes101|talk]]) 14:23, 27 April 2019 (PDT)<br /> <br /> So I thought about it a bit. The first fixed point of the elementary embedding $j$ above $crit(j)$ is the supremum of all of $crit(j)$, $j(crit(j))$, $j^2(crit(j))$, $j^3(crit(j))$, etc..., and said iterates of $j$ form a cofinal set of $j^\omega(crit(j))$. So for any $\alpha &lt; j^\omega(crit(j))$, there is some whole number $n$ such that $j^n(crit(j)) &gt; \alpha$. If I reasoned correctly, this means that $\omega$-almost hugeness is equivalent to $n$-hugeness for all whole number $n$, which is known to be perfectly consistent at least relative to an $I3$ cardinal. [[User:Eaglgenes101|Eaglgenes101]] ([[User talk:Eaglgenes101|talk]]) 08:26, 29 April 2019 (PDT)<br /> <br /> I thought a bit too. You've reasoned incorrectly, because while there could be a sequence of $M_n$ for each $n$ for which $j:V\rightarrow M_n$ witness the $n$-hugeness of a cardinal $\kappa$, there may not be a single model $M$ for which $j:V\rightarrow M$ witnesses the $n$-hugeness of $\kappa$ for all $n$. This would make a cardinal which is $n$-huge for all $n$ but not $\omega$-almost huge. [[User:Zetapology|Zetapology]] ([[User talk:Zetapology|talk]]) 11:05, 9 May 2019 (PST)</div> Zetapology http://cantorsattic.info/index.php?title=Vopenka&diff=2701 Vopenka 2018-12-04T22:56:32Z <p>Zetapology: /* Locally Presentable Categories */</p> <hr /> <div>{{DISPLAYTITLE: Vopěnka's principle and Vopěnka cardinals}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Reflection principles]]<br /> Vopěnka's principle is a large cardinal axiom at the upper end of the large cardinal hierarchy that is particularly notable for its applications to category theory. <br /> In a set theoretic setting, the most common definition is the following:<br /> &lt;blockquote&gt;<br /> For any language $\mathcal{L}$ and any proper class $C$ of $\mathcal{L}$-structures, there are distinct structures $M, N\in C$ and an [[elementary embedding]] $j:M\to N$.<br /> &lt;/blockquote&gt;<br /> For example, taking $\mathcal{L}$ to be the language with one unary and one binary predicate, we can consider for any ordinal $\eta$ the class of structures<br /> $\langle V_{\alpha+\eta},\{\alpha\},\in\rangle$, and conclude from Vopěnka's principle that a cardinal that is at least $\eta$-[[extendible]] exists.<br /> In fact if Vopěnka's principle holds then there are a proper class of extendible cardinals; bounding the strength of the axiom from above, we have that<br /> if $\kappa$ is [[huge#Almost huge|almost huge]], or even [[high-jump|almost-high-jump]], then $V_\kappa$ satisfies Vopěnka's principle. <br /> <br /> ==Formalizations==<br /> <br /> As stated above and from the point of view of ZFC, this is actually an axiom schema, as we quantify over proper classes, which from a purely ZFC perspective means definable proper classes. A somewhat stronger alternative is to view Vopěnka's principle as an axiom in second-order set theory capable to dealing with proper classes, such as von Neumann-Gödel-Bernays set theory. This is a strictly stronger assertion. [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably] Finally, one may relativize the principle to a particular cardinal, leading to the concept of a Vopěnka cardinal. <br /> <br /> == Vopěnka cardinals == <br /> <br /> An inaccessible cardinal $\kappa$ is a ''Vopěnka cardinal'' if and only if $V_\kappa$ satisfies Vopěnka's principle, that is, where we interpret the proper classes of $V_\kappa$ as the subsets of $V_\kappa$ of cardinality $\kappa$. Because of a characterization of Vopěnka's principle in terms of graphs, a cardinal $\kappa$ is Vopěnka if and only if $\kappa$ is inaccessible and any set $\kappa$-sized set $G$ of nonisomorphic graphs has some $g_0$ and $g_1$ with $g_0$ a proper subgraph of $g_1$. (Need to cite sources)<br /> <br /> Perlmutter &lt;cite&gt;Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge&lt;/cite&gt; proved that a cardinal is a Vopěnka cardinal if and only if it is a [[Woodin|Woodin for supercompactness]] cardinal. <br /> <br /> As we mentioned above, every almost huge cardinal is a Vopěnka cardinal.<br /> <br /> ==Equivalent statements==<br /> <br /> === $C^{(n)}$-extendible cardinals ===<br /> <br /> The schema form of Vopěnka's principle is equivalent to the existence of a proper class of $C^{(n)}$-[[extendible]] cardinals for every $n$; indeed there is a level-by-level stratification of Vopěnka's principle, with Vopěnka's principle for a $\Sigma_{n+2}$-definable class corresponds to the existence of a $C^{(n)}$-extendible cardinal greater than the ranks of the parameters.<br /> &lt;CITE&gt;BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses&lt;/CITE&gt;<br /> <br /> === Strong Compactness of Logics ===<br /> <br /> Vopěnka's principle is equivalent to the following statement about logics as well:<br /> <br /> For every logic $\mathcal{L}$, there is a cardinal $\mu_{\mathcal{L}}$ such that for any language $\tau$ and any $\mathcal{L}(\tau)$-theory $T$, $T$ is satisfiable if and only if every $t\subseteq T$ such that $|t|&lt;\mu_{\mathcal{L}}$ is satisfiable. &lt;cite&gt;Makowsky1985:CompactLogics&lt;/cite&gt;<br /> <br /> This $\mu_{\mathcal{L}}$ is called the strong compactness cardinal of $\mathcal{L}$. Vopěnka's principle therefore is equivalent to every logic having a strong compactness cardinal. This is very similar in definition to the Löwenheim–Skolem number of $\mathcal{L}$, although it is not guaranteed to exist. <br /> <br /> Here are some examples of strong compactness cardinals of specific logics:<br /> <br /> *If $\kappa\leq\lambda$ and $\lambda$ is [[strongly compact]] or $\aleph_0$, then the strong compactness cardinal of [[infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] is at most $\lambda$.<br /> *Similarly, if $\kappa\leq\lambda$ and $\lambda$ is [[extendible]], then for any natural number $n$, the strong compactness cardinal of $\mathcal{L}^n_{\kappa,\kappa}$ ($\mathcal{L}_{\kappa,\kappa}$ with $n+1$-th order logic) is at most $\lambda$. Therefore for any natural number $n$, the strong compactness cardinal of $n+1$-th order finitary logic is at most the least extendible cardinal.<br /> <br /> === Locally Presentable Categories ===<br /> <br /> Vopěnka's principle is equivalent to the axiom stating &quot;no large full subcategory $C$ of any locally presentable category is discrete.&quot; (Sources needed). Equivalently, no large full subcategory of Graph (the category of all graphs) is discrete; that is, for any proper class of simple directed graphs, there is at least one pair of nonequal graphs $G$ and $H$ in the class such that $G$ is a subgraph of $H$. This is a $\Pi^1_1$ statement, so the least Vopěnka cardinals are not even [[weakly compact]] (although the least weakly compact cardinal is much, much, much smaller than the least Vopěnka cardinal, if it exists).<br /> <br /> Intuitively, a &quot;category&quot; is just a class of mathematical objects with some notion of &quot;morphism&quot;, &quot;homomorphism&quot;, &quot;isomorphism&quot;, (etc.). For example, in Set, the category of all sets, homomorphisms are just injections, and isomorphisms are bijections. In categories of groups and models, homomorphisms and isomorphisms share their actual names.<br /> <br /> A &quot;locally small category&quot; $C$ is one with only set-many morphisms between any two objects of $C$. This is one where the objects of $C$ behave &quot;set-like&quot; in the sense that, usually, the number of morphisms between two set-sized objects is at most the number of functions between their universes (like in groups and in graphs). A &quot;locally presentable category&quot; is a locally small category with a couple more really nice properties; you can &quot;generate&quot; all of the objects from set-many objects in the category.<br /> <br /> Vopěnka's principle intuitively states that if you have a locally presentable category $C$, then any proper class of objects of $C$ has some nonisomorphic objects $c$ and $d$ where $c$ has a morphism into $d$.<br /> <br /> === Woodin cardinals ===<br /> <br /> There is a strange connection between the [[Woodin]] cardinals and the Vopěnka cardinals. In particular, Vopěnkaness is equivalent to two strengthening variants of Woodinness, namely the [[Woodin#Shelah|Woodin for Supercompactness]] cardinals and the [[n-fold Woodin|$2$-fold Woodin]] cardinals. As a result, every Vopěnka cardinal is Woodin.<br /> <br /> === Elementary Embeddings Between Ranks ===<br /> <br /> An equivalent statement to Vopěnka's principle is that for any proper class $C\subseteq ORD$, there are $\alpha\in C$, $\beta\in C$, and a nontrivial [[elementary embedding]] $j:\langle V_\alpha;\in,P\rangle\rightarrow\langle V_\beta;\in,P\rangle$. Vopěnka's principle quite obviously implies this. The reason the converse holds is because every elementary embedding can be &quot;encoded&quot; (in a sense) into one of these. For more information, see &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;.<br /> <br /> ==Other points to note==<br /> <br /> Whilst Vopěnka cardinals are very strong in terms of consistency strength, a Vopěnka cardinal need not even be [[weakly compact]]. Indeed, the definition of a Vopěnka cardinal is a $\Pi^1_1$ statement over $V_\kappa$ (Vopěnka's principle itself is $\Pi^1_1$), and [[indescribable|$\Pi^1_1$-indescribability]] is one of the equivalent definitions of weak compactness. Thus, the least weakly compact Vopěnka cardinal must have (many) other Vopěnka cardinals less than it.<br /> <br /> ==External links==<br /> <br /> * [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably Math Overflow question and answer about formalisations]<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Worldly&diff=2700 Worldly 2018-11-30T18:02:29Z <p>Zetapology: Added replacement characterization</p> <hr /> <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.<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.<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 $\text{ZF}^-$ ($\text{ZF}$ 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 $A\in V_\kappa$, $f&quot;A\in V_\kappa$ also.</div> Zetapology http://cantorsattic.info/index.php?title=Strongly_compact&diff=2691 Strongly compact 2018-11-02T18:02:50Z <p>Zetapology: /* Topological Relevance */</p> <hr /> <div>{{DISPLAYTITLE: Strongly compact cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an<br /> uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$. <br /> <br /> The strongly compact embedding characterizations are closely related to that of [[supercompact]] cardinals, which are characterized by [[elementary embedding|elementary embeddings]] with a high degree of closure: $\kappa$ is $\theta$-[[supercompact]] if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta&lt;j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-[[supercompact]] cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\subset M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.<br /> <br /> == Diverse characterizations ==<br /> <br /> There are diverse equivalent characterizations of the strongly compact cardinals. <br /> <br /> === Strong compactness characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha&lt;\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha&lt;\delta}\varphi_\alpha$ of any size $\delta&lt;\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\langle x_\alpha\mid\alpha&lt;\delta\rangle$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.<br /> <br /> === Strong compactness embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Cover property characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\subset M$ of size $\theta$ there is $s\in M$ with $t\subset s$ and $|s|^M&lt;j(\kappa)$.<br /> <br /> === Fine measure characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a [[filter|fine measure]] on $\mathcal{P}_\kappa(\theta)$. The notation $\mathcal{P}_\kappa(\theta)$ means $\{\sigma\subset\theta\mid |\sigma|&lt;\kappa\}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Filter extension characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete [[filter]] of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Discontinuous ultrapower characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda&lt;j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$. <br /> <br /> === Discontinuous embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda&lt;j(\lambda)$.<br /> <br /> === Ketonen characterization ===<br /> <br /> An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha&lt;\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality. <br /> <br /> === Regular ultrafilter characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\{X_\alpha\mid\alpha&lt;\theta\}\subset \mu$ such that $\bigcap_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.<br /> <br /> == Strongly compact cardinals and forcing ==<br /> <br /> If there is proper class-many strongly compact cardinals, then there is a [[forcing|generic model]] of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot;. If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot; + &quot;every uncountable cardinal is both almost [[Ramsey]] and a [[Rowbottom]] cardinal carrying a Rowbottom filter&quot;. <br /> This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals.<br /> <br /> == Relation to other large cardinal notions ==<br /> <br /> Strongly compact cardinals are [[measurable]]. The least strongly compact cardinal can be equal to the least measurable cardinal, or to the least [[supercompact]] cardinal, by results of Magidor. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)<br /> <br /> Even though strongly compact cardinals imply the consistency of the negation of the singular cardinal hypothesis, SCH, for any singular strong limit cardinal $\kappa$ above the least strongly compact cardinal, $2^\kappa=\kappa^+$ (also known as &quot;SCH holds above strong compactness&quot;). &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> If there is a strongly compact cardinal $\kappa$ then for all $\lambda\geq\kappa$ and $A\subseteq\lambda$, $\lambda^+$ is [[ineffable]] in $L[A]$.<br /> <br /> 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.<br /> <br /> Every strongly compact cardinal is [[strongly tall]], although the existence of a strongly compact cardinal is equiconsistent with &quot;the least measurable cardinal is the least strongly compact cardinal, and therefore the least strongly tall cardinal&quot;, so it could be the case that the least of the measurable, tall, strongly tall, and strongly compact cardinals all line up.<br /> <br /> == Topological Relevance ==<br /> <br /> Strongly compact cardinals are related to the topological notion of compactness, interestingly enough.<br /> <br /> === Intuition === <br /> <br /> A topological space $X$ is called $\kappa$-compact when every open cover has a subcover of size below $\kappa$. More intuitively, it &quot;looks&quot; as though it has size below $\kappa$. For example, the $\aleph_0$-compact subspaces of the real number line are just the subspaces which are bounded. For example, a shape with finite area could be considered $\aleph_0$-compact, even though the amount of points is not only infinite but continuum-sized.<br /> <br /> The product of a collection of spaces is a little difficult to describe intuitively. However, it notably increases the amount of &quot;dimensions&quot; to a space. For example, the product of $n$-copies of the real number line is just the $n$-dimensional euclidean space (the line, the plane, etc.). Also, the general intuition is that it doesn't make spaces any bigger than the biggest one in the collection, so the product of a bunch of small spaces and a big space should be no 'bigger' than the big space.<br /> <br /> The idea is that the product of $\kappa$-compact spaces should itself be $\kappa$-compact, since the product doesn't make spaces any &quot;bigger.&quot; However, there are examples of two $\aleph_1$-compact spaces (they &quot;look countably infinite&quot;) which combine to make a space which isn't $\aleph_1$-compact (&quot;looks uncountable&quot;). However, if $\kappa$ is strongly compact, then this intuition holds; the product of any $\kappa$-compact spaces is strongly compact. One could maybe see why strongly compact cardinals are so big then; they imply that combining a bunch of small-relative-to-$\kappa$ spaces together by adding arbitrarily many dimensions keeps the space looking small relative to $\kappa$.'<br /> <br /> Tychonoff's theorem is precisely the statement that the product of $\aleph_0$-compact spaces is $\aleph_0$-compact; that is, if you combine a bunch of finite-looking spaces together and keep adding more and more dimensions, you get a space which is finite-looking.<br /> <br /> ''(Sources to be added)''<br /> <br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Strongly_compact&diff=2690 Strongly compact 2018-10-31T06:09:32Z <p>Zetapology: added topological relevance</p> <hr /> <div>{{DISPLAYTITLE: Strongly compact cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an<br /> uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$. <br /> <br /> The strongly compact embedding characterizations are closely related to that of [[supercompact]] cardinals, which are characterized by [[elementary embedding|elementary embeddings]] with a high degree of closure: $\kappa$ is $\theta$-[[supercompact]] if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta&lt;j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-[[supercompact]] cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\subset M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.<br /> <br /> == Diverse characterizations ==<br /> <br /> There are diverse equivalent characterizations of the strongly compact cardinals. <br /> <br /> === Strong compactness characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha&lt;\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha&lt;\delta}\varphi_\alpha$ of any size $\delta&lt;\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\langle x_\alpha\mid\alpha&lt;\delta\rangle$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.<br /> <br /> === Strong compactness embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Cover property characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\subset M$ of size $\theta$ there is $s\in M$ with $t\subset s$ and $|s|^M&lt;j(\kappa)$.<br /> <br /> === Fine measure characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a [[filter|fine measure]] on $\mathcal{P}_\kappa(\theta)$. The notation $\mathcal{P}_\kappa(\theta)$ means $\{\sigma\subset\theta\mid |\sigma|&lt;\kappa\}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Filter extension characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete [[filter]] of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Discontinuous ultrapower characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda&lt;j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$. <br /> <br /> === Discontinuous embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda&lt;j(\lambda)$.<br /> <br /> === Ketonen characterization ===<br /> <br /> An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha&lt;\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality. <br /> <br /> === Regular ultrafilter characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\{X_\alpha\mid\alpha&lt;\theta\}\subset \mu$ such that $\bigcap_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.<br /> <br /> == Strongly compact cardinals and forcing ==<br /> <br /> If there is proper class-many strongly compact cardinals, then there is a [[forcing|generic model]] of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot;. If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot; + &quot;every uncountable cardinal is both almost [[Ramsey]] and a [[Rowbottom]] cardinal carrying a Rowbottom filter&quot;. <br /> This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals.<br /> <br /> == Relation to other large cardinal notions ==<br /> <br /> Strongly compact cardinals are [[measurable]]. The least strongly compact cardinal can be equal to the least measurable cardinal, or to the least [[supercompact]] cardinal, by results of Magidor. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)<br /> <br /> Even though strongly compact cardinals imply the consistency of the negation of the singular cardinal hypothesis, SCH, for any singular strong limit cardinal $\kappa$ above the least strongly compact cardinal, $2^\kappa=\kappa^+$ (also known as &quot;SCH holds above strong compactness&quot;). &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> If there is a strongly compact cardinal $\kappa$ then for all $\lambda\geq\kappa$ and $A\subseteq\lambda$, $\lambda^+$ is [[ineffable]] in $L[A]$.<br /> <br /> 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.<br /> <br /> Every strongly compact cardinal is [[strongly tall]], although the existence of a strongly compact cardinal is equiconsistent with &quot;the least measurable cardinal is the least strongly compact cardinal, and therefore the least strongly tall cardinal&quot;, so it could be the case that the least of the measurable, tall, strongly tall, and strongly compact cardinals all line up.<br /> <br /> == Topological Relevance ==<br /> <br /> Strongly compact cardinals are related to the topological notion of compactness, interestingly enough.<br /> <br /> === Specific Relations ===<br /> <br /> If $\kappa$ is uncountable, then the following are equivalent (under the [[axiom of choice]]):<br /> #$\kappa$ is strongly compact.<br /> #If $X$ is a space such that all $\kappa$-complete ultrafilters converge, then $X$ is $\kappa$-compact.<br /> #The product of any collection of $\kappa$-compact spaces is itself $\kappa$-compact.<br /> <br /> === Intuition === <br /> <br /> A topological space $X$ is called $\kappa$-compact when every open cover has a subcover of size below $\kappa$. More intuitively, it &quot;looks&quot; as though it has size below $\kappa$. For example, the $\aleph_0$-compact subspaces of the real number line are just the subspaces which are bounded. For example, a shape with finite area could be considered $\aleph_0$-compact, even though the amount of points is not only infinite but continuum-sized.<br /> <br /> The product of a collection of spaces is a little difficult to describe intuitively. However, it notably increases the amount of &quot;dimensions&quot; to a space. For example, the product of $n$-copies of the real number line is just the $n$-dimensional euclidean space (the line, the plane, etc.). Also, the general intuition is that it doesn't make spaces any bigger than the biggest one in the collection, so the product of a bunch of small spaces and a big space should be no 'bigger' than the big space.<br /> <br /> The idea is that the product of $\kappa$-compact spaces should itself be $\kappa$-compact, since the product doesn't make spaces any &quot;bigger.&quot; However, there are examples of two $\aleph_1$-compact spaces (they &quot;look countably infinite&quot;) which combine to make a space which isn't $\aleph_1$-compact (&quot;looks uncountable&quot;). However, if $\kappa$ is strongly compact, then this intuition holds; the product of any $\kappa$-compact spaces is strongly compact. One could maybe see why strongly compact cardinals are so big then; they imply that combining a bunch of small-relative-to-$\kappa$ spaces together by adding arbitrarily many dimensions keeps the space looking small relative to $\kappa$.'<br /> <br /> Tychonoff's theorem is precisely the statement that the product of $\aleph_0$-compact spaces is $\aleph_0$-compact; that is, if you combine a bunch of finite-looking spaces together and keep adding more and more dimensions, you get a space which is finite-looking.<br /> <br /> ''(Sources to be added)''<br /> <br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Partition_property&diff=2689 Partition property 2018-10-24T16:37:22Z <p>Zetapology: /* Theorems and Large Cardinal Axioms */</p> <hr /> <div>[[Category:Partition property]]<br /> A partition property is an infinitary combinatorical principle in set theory. Partition properties are best associated with various [[upper attic|large cardinal axioms]] (all of which are below [[measurable]]), but can also be associated with infinite graphs.<br /> <br /> The '''pigeonhole principle''' famously states that if there are $n$ pigeons in $n-m$ holes, then at least one hole contains at least $m$ pigeons. Partition properties are best motivated as generalizations of the pigeonhole principle to infinite cardinals. For example, if there are $\aleph_1$ pigeons and there are $\aleph_0$ many holes, then at least one hole contains $\aleph_1$ pigeons.<br /> <br /> == Definitions ==<br /> <br /> There are quite a few definitions involved with partition properties. In fact, partition calculus, the study of partition properties, almost completely either comprisse or describes most of infinitary combinatorics itself, so it would make sense that the terminology involved with it is quite unique.<br /> <br /> === Square Bracket Notation ===<br /> <br /> The square bracket notation is somewhat simple and easy to grasp (and used in many other places). Let $X$ be a set of ordinals. $[X]^\beta$ for some ordinal $\beta$ is the set of all subsets $x\subseteq X$ such that $(x,&lt;)$ has order-type $\beta$; that is, there is a bijection $f$ from $x$ to $\beta$ such that $f(a)&lt;f(b)$ iff $a&lt;b$ for each $a$ and $b$ in $x$. Such a bijection is often called an order-isomorphism.<br /> <br /> $[X]^{&lt;\beta}$ for some ordinal $\beta$ is simply defined as the union of all $[X]^{\alpha}$ for $\alpha&lt;\beta$, the set of all subsets $x\subseteq X$ with order-type less than $\beta$. In the case of $\omega$, $[X]^{&lt;\omega}$ is the set of all finite subsets of $X$.<br /> <br /> === Homogeneous Sets ===<br /> <br /> Let $f:[\kappa]^\beta\rightarrow\lambda$ be a function (in this study, such functions are often called partitions). A set $H\subseteq\kappa$ is then called '''homogeneous for $f$''' when for any two subsets $h_0,h_1\subseteq H$ of order type $\beta$, $f(h_0)=f(h_1)$. This is equivalent to $f$ being constant on $[H]^\beta$.<br /> <br /> In another case, let $f:[\kappa]^{&lt;\omega}\rightarrow\lambda$ be a function. A set $H\subseteq\kappa$ is then called '''homogeneous for $f$''' when for any two finite subsets $h_0,h_1\subseteq H$ of the same size, $f(h_0)=f(h_1)$. <br /> <br /> === The Various Partition Properties ===<br /> <br /> Let $\kappa$ and $\lambda$ be cardinals and let $\alpha$ and $\beta$ be ordinals. Then, the following notations are used for the partition properties:<br /> <br /> *$\kappa\rightarrow (\alpha)_\lambda^\beta$ iff for every function $f:[\kappa]^\beta\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ which is homogeneous for $f$. If $\alpha$ is a cardinal (which it most often is), then the requirement on $H$ can be loosened to $H$ having cardinality $\alpha$ and being homogeneous for $f$ without loss of generality. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *A common abbreviation for $\kappa\rightarrow (\alpha)_2^n$ is $\kappa\rightarrow (\alpha)^n$.<br /> *$\kappa\rightarrow (\alpha)_\lambda^{&lt;\omega}$ iff for every function $f:[\kappa]^{&lt;\omega}\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ which is homogeneous for $f$. If $\alpha$ is a cardinal (which it most often is), then the requirement on $H$ can be loosened to $H$ having cardinality $\alpha$ and being homogeneous for $f$ without loss of generality. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> Let $\nu$ be a cardinal. The '''square bracket partition properties''' are defined as follows:<br /> <br /> *$\kappa\rightarrow [\alpha]_\lambda^\beta$ iff for every function $f:[\kappa]^\beta\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ and an ordinal $\gamma&lt;\lambda$ such that $f(h)\neq\gamma$ for any $h\in [H]^\beta$.<br /> *$\kappa\rightarrow [\alpha]_\lambda^{&lt;\omega}$ iff for every function $f:[\kappa]^{&lt;\omega}\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ and an ordinal $\gamma&lt;\lambda$ such that $f(h)\neq\gamma$ for any finite subset $h$ of $H$. <br /> *$\kappa\rightarrow [\alpha]_{\lambda,&lt;\nu}^\beta$ iff for every function $f:[\kappa]^\beta\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ such that $f$ restricted to $[H]^\beta$ yields less than $\nu$-many distinct outputs. Note that $\kappa\rightarrow[\alpha]_{\lambda,&lt;2}^\beta$ iff $\kappa\rightarrow(\alpha)_\lambda^\beta$.<br /> *$\kappa\rightarrow [\alpha]_{\lambda,&lt;\nu}^{&lt;\omega}$ iff for every function $f:[\kappa]^{&lt;\omega}\rightarrow\lambda$, there is set $H$ of order-type $\alpha$ such that $f$ restricted to $[H]^{&lt;\omega}$ yields less than $\nu$-many distinct outputs.<br /> <br /> == Theorems and Large Cardinal Axioms ==<br /> <br /> There are several theorems in the study of partition calculus. Namely:<br /> <br /> *Ramsey's theorem, which states that $\aleph_0\rightarrow (\omega)_m^n$ for each finite $m$ and $n$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> *$2^\kappa\not\rightarrow (\kappa^+)^2$ &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> *The Erdős-Rado theorem, which states that $\beth_n(\kappa)^+\rightarrow (\kappa^+)_\kappa^{n+1}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *$\kappa\not\rightarrow(\omega)_2^\omega$ &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *For any finite nonzero $n$ and ordinals $\alpha$ and $\beta$, there is a $\kappa$ such that $\kappa\rightarrow(\alpha)_\beta^n$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *The Gödel-Erdős-Kakutani theorem, which states that $2^\kappa\not\rightarrow (3)^2_\kappa$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *$\kappa\not\rightarrow [\kappa]_\kappa^\omega$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *$\lambda^+\not\rightarrow[\lambda+1]^2_{\lambda,&lt;\lambda}$ &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *$\lambda\not\rightarrow[\lambda]^1_{\mathrm{cf}(\lambda),&lt;\mathrm{cf}(\lambda)}$ &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *For any regular $\kappa$, $\kappa^+\not\rightarrow[\kappa^+]^2_{\kappa^+}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *For any [[inaccessible]] cardinal $\kappa$, $\kappa\rightarrow(\lambda)_\lambda^2$ for any $\lambda&lt;\kappa$ &lt;cite&gt;Drake1974:SetTheory&lt;/cite&gt; . However, this may not be an equivalence; if the [[continuum hypothesis]] holds at $\kappa$, then $(\kappa^{++})\rightarrow(\lambda)^2_\kappa$ for any $\lambda&lt;\kappa^{++}$.<br /> <br /> In terms of large cardinal axioms, many can be described using a partition property. Here are those which can be found on this website:<br /> <br /> *Although not a large cardinal itself, [[Chang's conjecture]] holds iff $\aleph_2\rightarrow[\omega_1]^{&lt;\omega}_{\aleph_1,&lt;\aleph_1}$, iff $\aleph_2\rightarrow[\omega_1]^{n}_{\aleph_1,&lt;\aleph_1}$ for some $n$, iff $\aleph_2\rightarrow[\omega_1]^{n}_{\aleph_1,&lt;\aleph_1}$ for every finite $n$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *A cardinal $\kappa$ is [[Ramsey]] iff $\kappa\rightarrow(\kappa)_\lambda^{&lt;\omega}$ for some $\lambda&gt;1$, iff $\kappa\rightarrow(\kappa)_\lambda^{&lt;\omega}$ for every $\lambda&lt;\kappa$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;&lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> *A cardinal $\kappa$ is the [[Erdos|$\alpha$-Erdős]] cardinal iff it is the smallest $\kappa$ such that $\kappa\rightarrow(\alpha)^{&lt;\omega}$.<br /> *A cardinal $\kappa$ is defined to be [[Rowbottom|$\nu$-Rowbottom]] iff $\kappa\rightarrow[\kappa]_{\lambda,&lt;\nu}^{&lt;\omega}$ for every $\lambda&lt;\kappa$.<br /> *A cardinal $\kappa$ is [[Jonsson|Jónsson]] iff $\kappa\rightarrow[\kappa]_\kappa^{&lt;\omega}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *A cardinal $\kappa$ is [[weakly compact]] iff $\kappa\rightarrow(\kappa)^2_\lambda$ for some $\lambda&gt;1$, iff $\kappa\rightarrow(\kappa)^2_\lambda$ for every $\lambda&lt;\kappa$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Library&diff=2688 Library 2018-10-24T16:29:46Z <p>Zetapology: fixed author</p> <hr /> <div>{{DISPLAYTITLE: The Cantor's attic library}}<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 /> &lt;!-- <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 /> //--&gt;<br /> <br /> &lt;biblio force=true&gt;<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 (04A20)},<br /> MRNUMBER = {0460120 (57 \#116)},<br /> MRREVIEWER = {Thomas J. Jech}<br /> }<br /> <br /> #BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@unpublished {BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br /> AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosicky, Jirí},<br /> TITLE = &quot;Definable orthogonality classes in accessible categories are small&quot;,<br /> NOTE = &quot;submitted for publication&quot;,<br /> url = {http://arxiv.org/abs/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 /> url = {http://jdh.hamkins.org/superstrong-never-indestructible/}<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 (Colloq., Keszthely, 1973; dedicated to P. Erd&amp;#337;s on his 60th birthday), Vol. I},<br /> PAGES = {109--130. Colloq. Math. Soc. J&amp;#225;nos Bolyai, Vol. 10},<br /> PUBLISHER = {North-Holland},<br /> ADDRESS = {Amsterdam},<br /> YEAR = {1975},<br /> MRCLASS = {02K35 (04A20)},<br /> MRNUMBER = {0384553 (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 = &quot;Blass, Andreas&quot;,<br /> fjournal = &quot;Pacific Journal of Mathematics&quot;,<br /> journal = &quot;Pacific J. Math.&quot;,<br /> number = &quot;2&quot;,<br /> pages = &quot;335--346&quot;,<br /> publisher = &quot;Pacific Journal of Mathematics, A Non-profit Corporation&quot;,<br /> title = &quot;Exact functors and measurable cardinals.&quot;,<br /> url = &quot;https://projecteuclid.org:443/euclid.pjm/1102867389&quot;,<br /> volume = &quot;63&quot;,<br /> year = &quot;1976&quot;<br /> }<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 /> url = {https://arxiv.org/pdf/1708.07561.pdf},<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 /> #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 /> url = {https://arxiv.org/pdf/1305.5961.pdf},<br /> }<br /> <br /> #CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br /> AUTHOR= {Cody, Brent and Gitman, Victoria},<br /> TITLE= {Easton's theorem for Ramsey and strongly Ramsey cardinals},<br /> NOTE= {In preparation}}<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:WholenessAxiom,<br /> AUTHOR = {Corazza, Paul},<br /> TITLE = {The gap between ${\rm I}_3$ 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 (03E65)},<br /> MRNUMBER = {MR2028926 (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 /> #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 (03C62 03E35)},<br /> MRNUMBER = {611394 (82i:03063)},<br /> MRREVIEWER = {F. R. 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Hungar.},<br /> FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br /> VOLUME = {13},<br /> YEAR = {1962},<br /> PAGES = {223--226},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.60},<br /> MRNUMBER = {0141603 (25 \#5001)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br /> AUTHOR = {Erd&amp;#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 /> VOLUME = {9},<br /> YEAR = {1958},<br /> PAGES = {111--131},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.00},<br /> MRNUMBER = {0095124 (20 \#1630)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #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 /> url = {https://arxiv.org/abs/1607.04904v4}<br /> eprint = {arXiv:1607.04904},<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 /> doi = {10.1002/malq.19960420109},<br /> volume = {42},<br /> 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 /> journal = {Mathematical Logic Quarterly},<br /> doi = {10.1002/malq.19970430309},<br /> volume = {43},<br /> pages = {369--377},<br /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19970430309/abstract}<br /> }<br /> <br /> #Esser99:ConsistencyPositiveTheory bibtex=@article{Esser96:ConsistencyPositiveTheory,<br /> author = {Esser, Olivier},<br /> 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 /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19990450110/abstract}<br /> }<br /> <br /> #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 /> volume = {65},<br /> number = {4},<br /> pages = {1911--1916},<br /> doi = {10.2307/2695086},<br /> url = {http://www.jstor.org/stable/2695086}<br /> }<br /> <br /> #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 /> volume = {19},<br /> pages = {97--100},<br /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.200310009/abstract}<br /> }<br /> <br /> #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 /> #Foreman2010:Handbook bibtex=@book<br /> {Foreman2010:Handbook, <br /> 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 /> <br /> #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. 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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 /> #GitmanJohnstone:IndestructiblyRamsey bibtex=@ARTICLE{GitmanJohnstone:IndestructiblyRamsey,<br /> AUTHOR= {Gitman, Victoria and Johnstone, Thomas},<br /> TITLE= {Indestructibility for Ramsey and Ramsey-like cardinals},<br /> NOTE= {In preparation}}<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. 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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 (03E65)},<br /> MRNUMBER = {1816602 (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 /> #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 (03E35)},<br /> MRNUMBER = {2489293 (2010g:03083)},<br /> MRREVIEWER = {Carlos A. Di Prisco},<br /> DOI = {10.1002/malq.200710084},<br /> URL = {http://boolesrings.org/hamkins/tallcardinals/},<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 /> url={https://arxiv.org/pdf/math/9909029.pdf/},<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 /> 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 /> url = {https://arxiv.org/pdf/1403.2788.pdf},<br /> eprint = {1403.2788},<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(83)90020-9},<br /> url={https://ac.els-cdn.com/0168007283900209/1-s2.0-0168007283900209-main.pdf?_tid=466bc36a-c95e-11e7-bf33-00000aab0f27&amp;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 /> doi = {10.1016/0168-0072(89)90030-4},<br /> url={https://ac.els-cdn.com/0168007289900304/1-s2.0-0168007289900304-main.pdf?_tid=2f5a4ffe-e130-11e7-9794-00000aacb361&amp;acdnat=1513298453_24fe48742f365da91523f1174bb74117}<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 /> <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 $L(\mathbb{R})$},<br /> YEAR = {2015},<br /> DOI = {10.1017/jsl.2014.49},<br /> URL = {https://arxiv.org/abs/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 (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 (80b:03083)},<br /> MRREVIEWER = {J. L. Bell},<br /> url = {http://math.bu.edu/people/aki/e.pdf},<br /> }<br /> <br /> <br /> #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 /> YEAR = {2009},<br /> PAGES = {xxii+536},<br /> URL = {https://link.springer.com/book/10.1007%2F978-3-540-88867-3}<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'''(1978)}, <br /> year = {1978}, <br /> url = {http://math.bu.edu/people/aki/d.pdf},}<br /> <br /> #Kentaro2007:DoubleHelix bibtex=@article{Kentaro2007:DoubleHelix,<br /> AUTHOR = {Kentaro, Sato},<br /> TITLE = {Double helix in large large cardinals and iteration of<br /> elementary embeddings},<br /> SERIES = {Annals of Pure and Applied Logic},<br /> PUBLISHER = {Elsevier B.V.},<br /> YEAR = {2007},<br /> URL = {https://ac.els-cdn.com/S0168007207000127/1-s2.0-S0168007207000127-main.pdf?_tid=aa889390-c1e4-11e7-a507-00000aacb362&amp;acdnat=1509857531_549949bbb11277bb53825de297d7dc00},<br /> }<br /> <br /> #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 /> VOLUME = {43},<br /> NUMBER = {1},<br /> PAGES = {65--76},<br /> URL = {http://www.jstor.org/stable/2271949}<br /> }<br /> <br /> #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 /> editor = {Foreman, Mathew; Kanamori, Akihiro},<br /> year = {2010},<br /> publisher = {Springer},<br /> url = {http://logic.harvard.edu/koellner/LCFD.pdf}<br /> }<br /> <br /> #Larson2010:HistoryDeterminacy bibtex=@article{<br /> {Larson2010:HistoryDeterminacy,<br /> AUTHOR = {Larson, Paul B.},<br /> TITLE = {A brief history of determinacy},<br /> YEAR = {2013},<br /> URL = {http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf}<br /> }<br /> <br /> #Laver1997:Implications bibtex=@article {Laver1997:Implications,<br /> AUTHOR = {Laver, Richard},<br /> TITLE = {Implications between strong large cardinal axioms},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {90},<br /> YEAR = {1997},<br /> NUMBER = {1--3},<br /> PAGES = {79--90},<br /> ISSN = {0168-0072},<br /> MRCLASS = {03E55 (03E35)},<br /> MRNUMBER = {1489305 (99c:03074)},<br /> MRREVIEWER = {Douglas R. Burke},<br /> }<br /> <br /> #Maddy88:BelAxiomsI bibtex=@article{Maddy88:BelAxiomsI,<br /> AUTHOR = {Maddy, Penelope},<br /> TITLE = {Believing the axioms. I},<br /> JOURNAL = {J. Symbolic Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {2},<br /> PAGES = {181--511},<br /> DOI = {10.2307/2274520},<br /> URL = {http://www.jstor.org/stable/2274520}<br /> }<br /> #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 /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {3},<br /> PAGES = {736--764},<br /> DOI = {10.2307/2274569},<br /> URL = {http://www.jstor.org/stable/2274569}<br /> }<br /> #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 /> }<br /> <br /> #Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br /> AUTHOR = {Makowsky, Johann},<br /> TITLE = {Vopěnka's Principle and Compact Logics},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents<br /> YEAR = {1985},<br /> }<br /> <br /> #Mitchell1997:JonssonErdosCoreModel bibtex=@article{#Mitchell1997:JonssonErdosCoreModel,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {Jónsson Cardinals, Erdős Cardinals, and the Core Model},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = {https://arxiv.org/pdf/math/9706207.pdf},<br /> YEAR = {1997}<br /> }<br /> <br /> #Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {The Covering Lemma},<br /> JOURNAL = {Handbook of Set Theory},<br /> EDITOR = {M. Foreman and A. Kanamori and M. Magidor},<br /> URL = {http://www.math.cas.cz/~jech/library/mitchell/covering.ps},<br /> YEAR = {2001}<br /> }<br /> <br /> #Miyamoto1998:ANoteOnWeakSegmentsOfPFA bibtex=@article{Miyamoto1998:ANoteOnWeakSegmentsOfPFA,<br /> AUTHOR = {Miyamoto, Tadatoshi}.<br /> TITLE = {A note on weak segments of PFA},<br /> JOURNAL = {Proceedings of the sixth Asian logic conference},<br /> YEAR = {1998},<br /> PAGES = {175--197}<br /> }<br /> <br /> #Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge bibtex=@article{Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge,<br /> AUTHOR = {Perlmutter, Norman}.<br /> TITLE = {The large cardinals between supercompact and almost-huge},<br /> YEAR = {2010},<br /> URL = {https://arxiv.org/pdf/1307.7387.pdf},<br /> }<br /> <br /> <br /> #Rathjen2006:OrdinalAnalysis bibtex=@article{Rathjen2006:OrdinalAnalysis,<br /> AUTHOR = {Rathjen, Michael}.<br /> TITLE = {The art of ordinal analysis},<br /> YEAR = {2006},<br /> URL = {http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf}<br /> }<br /> <br /> #SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br /> AUTHOR = {Sharpe, Ian and Welch, Philip},<br /> TITLE = {Greatly Erd&amp;#337;s cardinals with some generalizations to<br /> the Chang and Ramsey properties},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {162},<br /> YEAR = {2011},<br /> NUMBER = {11},<br /> PAGES = {863--902},<br /> ISSN = {0168-0072},<br /> CODEN = {APALD7},<br /> MRCLASS = {03E04 (03E35 03E45 03E55)},<br /> MRNUMBER = {2817562},<br /> DOI = {10.1016/j.apal.2011.04.002},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2011.04.002},<br /> }<br /> <br /> #Shelah1994:CardinalArithmetic bibtex=@article {#Shelah1994:CardinalArithmetic,<br /> AUTHOR = {Shelah, Saharon},<br /> TITLE = {Cardinal Arithmetic},<br /> JOURNAL = {Oxford Logic Guides},<br /> VOLUME = {29},<br /> YEAR = {1994},<br /> }<br /> <br /> #Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Partial near supercompactness},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> DOI = {10.1016/j.apal.2012.08.001},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2012.08.001},<br /> YEAR = {2012},<br /> NOTE = {In Press.}<br /> }<br /> <br /> #Schanker2011:WeaklyMeasurableCardinals bibtex=@article{Schanker2011:WeaklyMeasurableCardinals,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals},<br /> YEAR = {2011},<br /> JOURNAL = {MLQ Math. Log. Q.},<br /> FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br /> VOLUME = {57},<br /> NUMBER = {3},<br /> PAGES = {266--280},<br /> DOI = {10.1002/malq.201010006},<br /> URL = {http://dx.doi.org/10.1002/malq.201010006}<br /> }<br /> <br /> #Schanker2011:Thesis bibtex=@phdthesis{Schanker2011:Thesis,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals and partial near supercompactness},<br /> SCHOOL = {CUNY Graduate Center},<br /> YEAR = {2011}<br /> }<br /> #Schindler2000:RemarkableCardinal bibtex=@article {Schindler2000:RemarkableCardinal,<br /> AUTHOR = {Schindler, Ralf-Dieter},<br /> TITLE = {Proper forcing and remarkable cardinals},<br /> JOURNAL = {Bull. Symbolic Logic},<br /> FJOURNAL = {The Bulletin of Symbolic Logic},<br /> VOLUME = {6},<br /> YEAR = {2000},<br /> NUMBER = {2},<br /> PAGES = {176--184},<br /> ISSN = {1079-8986},<br /> MRCLASS = {03E40 (03E45 03E55)},<br /> MRNUMBER = {1765054 (2001h:03096)},<br /> MRREVIEWER = {A. Kanamori},<br /> DOI = {10.2307/421205},<br /> URL = {http://dx.doi.org/10.2307/421205},<br /> }<br /> #Silver1970:ErdosCardinal bibtex=@article {MR0274278,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {A large cardinal in the constructible universe},<br /> JOURNAL = {Fund. Math.},<br /> FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br /> VOLUME = {69},<br /> YEAR = {1970},<br /> PAGES = {93--100},<br /> ISSN = {0016-2736},<br /> MRCLASS = {02.65},<br /> MRNUMBER = {0274278 (43 \#43)},<br /> MRREVIEWER = {N. C. A. da Costa},<br /> }<br /> #Silver1971:ZeroSharp bibtex=@article {MR0409188,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {Some applications of model theory in set theory},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {3},<br /> YEAR = {1971},<br /> NUMBER = {1},<br /> PAGES = {45--110},<br /> ISSN = {0168-0072},<br /> MRCLASS = {02K35},<br /> MRNUMBER = {0409188 (53 \#12950)},<br /> MRREVIEWER = {Andreas Blass},<br /> }<br /> #Suzuki1998:NojVtoVinVofG bibtex=@article{Suzuki1998:NojVtoVinV[G],<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 (03E05)},<br /> MRNUMBER = {MR1650737 (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 (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 /> URL = {https://arxiv.org/abs/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 /> #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 (03E05 03E55)},<br /> MRNUMBER = {MR2838054 (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 /> URL = {https://arxiv.org/pdf/math/9611209.pdf},<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&amp;rep=rep1&amp;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 /> #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 /> <br /> URL = {http://www.worldscientific.com/doi/abs/10.1142/S021906131000095X},<br /> eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br /> }<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 (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 /> <br /> &lt;/biblio&gt;<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> Zetapology http://cantorsattic.info/index.php?title=Library&diff=2687 Library 2018-10-24T16:28:40Z <p>Zetapology: </p> <hr /> <div>{{DISPLAYTITLE: The Cantor's attic library}}<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 /> &lt;!-- <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 /> //--&gt;<br /> <br /> &lt;biblio force=true&gt;<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 (04A20)},<br /> MRNUMBER = {0460120 (57 \#116)},<br /> MRREVIEWER = {Thomas J. Jech}<br /> }<br /> <br /> #BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@unpublished {BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br /> AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosicky, Jirí},<br /> TITLE = &quot;Definable orthogonality classes in accessible categories are small&quot;,<br /> NOTE = &quot;submitted for publication&quot;,<br /> url = {http://arxiv.org/abs/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 /> url = {http://jdh.hamkins.org/superstrong-never-indestructible/}<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 (Colloq., Keszthely, 1973; dedicated to P. Erd&amp;#337;s on his 60th birthday), Vol. I},<br /> PAGES = {109--130. Colloq. Math. Soc. J&amp;#225;nos Bolyai, Vol. 10},<br /> PUBLISHER = {North-Holland},<br /> ADDRESS = {Amsterdam},<br /> YEAR = {1975},<br /> MRCLASS = {02K35 (04A20)},<br /> MRNUMBER = {0384553 (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 = &quot;Blass, Andreas&quot;,<br /> fjournal = &quot;Pacific Journal of Mathematics&quot;,<br /> journal = &quot;Pacific J. 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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 (03E65)},<br /> MRNUMBER = {MR2028926 (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 /> #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 (03C62 03E35)},<br /> MRNUMBER = {611394 (82i:03063)},<br /> MRREVIEWER = {F. R. 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Hungar.},<br /> FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br /> VOLUME = {13},<br /> YEAR = {1962},<br /> PAGES = {223--226},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.60},<br /> MRNUMBER = {0141603 (25 \#5001)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br /> AUTHOR = {Erd&amp;#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 /> VOLUME = {9},<br /> YEAR = {1958},<br /> PAGES = {111--131},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.00},<br /> MRNUMBER = {0095124 (20 \#1630)},<br /> MRREVIEWER = {L. 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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 /> #Foreman2010:Handbook bibtex=@book<br /> {Foreman2010:Handbook, <br /> 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 /> <br /> #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. 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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 (03E65)},<br /> MRNUMBER = {1816602 (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 /> #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 (03E35)},<br /> MRNUMBER = {2489293 (2010g:03083)},<br /> MRREVIEWER = {Carlos A. 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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 (80b:03083)},<br /> MRREVIEWER = {J. L. Bell},<br /> url = {http://math.bu.edu/people/aki/e.pdf},<br /> }<br /> <br /> <br /> #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 /> YEAR = {2009},<br /> PAGES = {xxii+536},<br /> URL = {https://link.springer.com/book/10.1007%2F978-3-540-88867-3}<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'''(1978)}, <br /> year = {1978}, <br /> url = {http://math.bu.edu/people/aki/d.pdf},}<br /> <br /> #Kentaro2007:DoubleHelix bibtex=@article{Kentaro2007:DoubleHelix,<br /> AUTHOR = {Kentaro, Sato},<br /> TITLE = {Double helix in large large cardinals and iteration of<br /> elementary embeddings},<br /> SERIES = {Annals of Pure and Applied Logic},<br /> PUBLISHER = {Elsevier B.V.},<br /> YEAR = {2007},<br /> URL = {https://ac.els-cdn.com/S0168007207000127/1-s2.0-S0168007207000127-main.pdf?_tid=aa889390-c1e4-11e7-a507-00000aacb362&amp;acdnat=1509857531_549949bbb11277bb53825de297d7dc00},<br /> }<br /> <br /> #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 /> VOLUME = {43},<br /> NUMBER = {1},<br /> PAGES = {65--76},<br /> URL = {http://www.jstor.org/stable/2271949}<br /> }<br /> <br /> #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 /> editor = {Foreman, Mathew; Kanamori, Akihiro},<br /> year = {2010},<br /> publisher = {Springer},<br /> url = {http://logic.harvard.edu/koellner/LCFD.pdf}<br /> }<br /> <br /> #Larson2010:HistoryDeterminacy bibtex=@article{<br /> {Larson2010:HistoryDeterminacy,<br /> AUTHOR = {Larson, Paul B.},<br /> TITLE = {A brief history of determinacy},<br /> YEAR = {2013},<br /> URL = {http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf}<br /> }<br /> <br /> #Laver1997:Implications bibtex=@article {Laver1997:Implications,<br /> AUTHOR = {Laver, Richard},<br /> TITLE = {Implications between strong large cardinal axioms},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {90},<br /> YEAR = {1997},<br /> NUMBER = {1--3},<br /> PAGES = {79--90},<br /> ISSN = {0168-0072},<br /> MRCLASS = {03E55 (03E35)},<br /> MRNUMBER = {1489305 (99c:03074)},<br /> MRREVIEWER = {Douglas R. Burke},<br /> }<br /> <br /> #Maddy88:BelAxiomsI bibtex=@article{Maddy88:BelAxiomsI,<br /> AUTHOR = {Maddy, Penelope},<br /> TITLE = {Believing the axioms. I},<br /> JOURNAL = {J. Symbolic Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {2},<br /> PAGES = {181--511},<br /> DOI = {10.2307/2274520},<br /> URL = {http://www.jstor.org/stable/2274520}<br /> }<br /> #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 /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {3},<br /> PAGES = {736--764},<br /> DOI = {10.2307/2274569},<br /> URL = {http://www.jstor.org/stable/2274569}<br /> }<br /> #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 /> }<br /> <br /> #Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br /> AUTHOR = {Makowsky, Johann},<br /> TITLE = {Vopěnka's Principle and Compact Logics},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents<br /> YEAR = {1985},<br /> }<br /> <br /> #Mitchell1997:JonssonErdosCoreModel bibtex=@article{#Mitchell1997:JonssonErdosCoreModel,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {Jónsson Cardinals, Erdős Cardinals, and the Core Model},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = {https://arxiv.org/pdf/math/9706207.pdf},<br /> YEAR = {1997}<br /> }<br /> <br /> #Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {The Covering Lemma},<br /> JOURNAL = {Handbook of Set Theory},<br /> EDITOR = {M. Foreman and A. Kanamori and M. Magidor},<br /> URL = {http://www.math.cas.cz/~jech/library/mitchell/covering.ps},<br /> YEAR = {2001}<br /> }<br /> <br /> #Miyamoto1998:ANoteOnWeakSegmentsOfPFA bibtex=@article{Miyamoto1998:ANoteOnWeakSegmentsOfPFA,<br /> AUTHOR = {Miyamoto, Tadatoshi}.<br /> TITLE = {A note on weak segments of PFA},<br /> JOURNAL = {Proceedings of the sixth Asian logic conference},<br /> YEAR = {1998},<br /> PAGES = {175--197}<br /> }<br /> <br /> #Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge bibtex=@article{Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge,<br /> AUTHOR = {Perlmutter, Norman}.<br /> TITLE = {The large cardinals between supercompact and almost-huge},<br /> YEAR = {2010},<br /> URL = {https://arxiv.org/pdf/1307.7387.pdf},<br /> }<br /> <br /> <br /> #Rathjen2006:OrdinalAnalysis bibtex=@article{Rathjen2006:OrdinalAnalysis,<br /> AUTHOR = {Rathjen, Michael}.<br /> TITLE = {The art of ordinal analysis},<br /> YEAR = {2006},<br /> URL = {http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf}<br /> }<br /> <br /> #SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br /> AUTHOR = {Sharpe, Ian and Welch, Philip},<br /> TITLE = {Greatly Erd&amp;#337;s cardinals with some generalizations to<br /> the Chang and Ramsey properties},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {162},<br /> YEAR = {2011},<br /> NUMBER = {11},<br /> PAGES = {863--902},<br /> ISSN = {0168-0072},<br /> CODEN = {APALD7},<br /> MRCLASS = {03E04 (03E35 03E45 03E55)},<br /> MRNUMBER = {2817562},<br /> DOI = {10.1016/j.apal.2011.04.002},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2011.04.002},<br /> }<br /> <br /> #Shelah1994:CardinalArithmetic bibtex=@article {#Shelah1994:CardinalArithmetic,<br /> AUTHOR = {Shelah, Saharon},<br /> TITLE = {Cardinal Arithmetic},<br /> JOURNAL = {Oxford Logic Guides},<br /> VOLUME = {29},<br /> YEAR = {1994},<br /> }<br /> <br /> #Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Partial near supercompactness},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> DOI = {10.1016/j.apal.2012.08.001},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2012.08.001},<br /> YEAR = {2012},<br /> NOTE = {In Press.}<br /> }<br /> <br /> #Schanker2011:WeaklyMeasurableCardinals bibtex=@article{Schanker2011:WeaklyMeasurableCardinals,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals},<br /> YEAR = {2011},<br /> JOURNAL = {MLQ Math. Log. Q.},<br /> FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br /> VOLUME = {57},<br /> NUMBER = {3},<br /> PAGES = {266--280},<br /> DOI = {10.1002/malq.201010006},<br /> URL = {http://dx.doi.org/10.1002/malq.201010006}<br /> }<br /> <br /> #Schanker2011:Thesis bibtex=@phdthesis{Schanker2011:Thesis,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals and partial near supercompactness},<br /> SCHOOL = {CUNY Graduate Center},<br /> YEAR = {2011}<br /> }<br /> #Schindler2000:RemarkableCardinal bibtex=@article {Schindler2000:RemarkableCardinal,<br /> AUTHOR = {Schindler, Ralf-Dieter},<br /> TITLE = {Proper forcing and remarkable cardinals},<br /> JOURNAL = {Bull. Symbolic Logic},<br /> FJOURNAL = {The Bulletin of Symbolic Logic},<br /> VOLUME = {6},<br /> YEAR = {2000},<br /> NUMBER = {2},<br /> PAGES = {176--184},<br /> ISSN = {1079-8986},<br /> MRCLASS = {03E40 (03E45 03E55)},<br /> MRNUMBER = {1765054 (2001h:03096)},<br /> MRREVIEWER = {A. Kanamori},<br /> DOI = {10.2307/421205},<br /> URL = {http://dx.doi.org/10.2307/421205},<br /> }<br /> #Silver1970:ErdosCardinal bibtex=@article {MR0274278,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {A large cardinal in the constructible universe},<br /> JOURNAL = {Fund. Math.},<br /> FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br /> VOLUME = {69},<br /> YEAR = {1970},<br /> PAGES = {93--100},<br /> ISSN = {0016-2736},<br /> MRCLASS = {02.65},<br /> MRNUMBER = {0274278 (43 \#43)},<br /> MRREVIEWER = {N. C. A. da Costa},<br /> }<br /> #Silver1971:ZeroSharp bibtex=@article {MR0409188,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {Some applications of model theory in set theory},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {3},<br /> YEAR = {1971},<br /> NUMBER = {1},<br /> PAGES = {45--110},<br /> ISSN = {0168-0072},<br /> MRCLASS = {02K35},<br /> MRNUMBER = {0409188 (53 \#12950)},<br /> MRREVIEWER = {Andreas Blass},<br /> }<br /> #Suzuki1998:NojVtoVinVofG bibtex=@article{Suzuki1998:NojVtoVinV[G],<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 (03E05)},<br /> MRNUMBER = {MR1650737 (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 (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 /> URL = {https://arxiv.org/abs/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 /> #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 (03E05 03E55)},<br /> MRNUMBER = {MR2838054 (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 /> URL = {https://arxiv.org/pdf/math/9611209.pdf},<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&amp;rep=rep1&amp;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 /> #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 /> <br /> URL = {http://www.worldscientific.com/doi/abs/10.1142/S021906131000095X},<br /> eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br /> }<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 (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 /> <br /> &lt;/biblio&gt;<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> Zetapology http://cantorsattic.info/index.php?title=Library&diff=2686 Library 2018-10-24T16:27:20Z <p>Zetapology: </p> <hr /> <div>{{DISPLAYTITLE: The Cantor's attic library}}<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 /> &lt;!-- <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 /> //--&gt;<br /> <br /> &lt;biblio force=true&gt;<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 (04A20)},<br /> MRNUMBER = {0460120 (57 \#116)},<br /> MRREVIEWER = {Thomas J. Jech}<br /> }<br /> <br /> #BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@unpublished {BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br /> AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosicky, Jirí},<br /> TITLE = &quot;Definable orthogonality classes in accessible categories are small&quot;,<br /> NOTE = &quot;submitted for publication&quot;,<br /> url = {http://arxiv.org/abs/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 /> url = {http://jdh.hamkins.org/superstrong-never-indestructible/}<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 (Colloq., Keszthely, 1973; dedicated to P. Erd&amp;#337;s on his 60th birthday), Vol. I},<br /> PAGES = {109--130. Colloq. Math. Soc. J&amp;#225;nos Bolyai, Vol. 10},<br /> PUBLISHER = {North-Holland},<br /> ADDRESS = {Amsterdam},<br /> YEAR = {1975},<br /> MRCLASS = {02K35 (04A20)},<br /> MRNUMBER = {0384553 (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 = &quot;Blass, Andreas&quot;,<br /> fjournal = &quot;Pacific Journal of Mathematics&quot;,<br /> journal = &quot;Pacific J. Math.&quot;,<br /> number = &quot;2&quot;,<br /> pages = &quot;335--346&quot;,<br /> publisher = &quot;Pacific Journal of Mathematics, A Non-profit Corporation&quot;,<br /> title = &quot;Exact functors and measurable cardinals.&quot;,<br /> url = &quot;https://projecteuclid.org:443/euclid.pjm/1102867389&quot;,<br /> volume = &quot;63&quot;,<br /> year = &quot;1976&quot;<br /> }<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 /> url = {https://arxiv.org/pdf/1708.07561.pdf},<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 /> #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 /> url = {https://arxiv.org/pdf/1305.5961.pdf},<br /> }<br /> <br /> #CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br /> AUTHOR= {Cody, Brent and Gitman, Victoria},<br /> TITLE= {Easton's theorem for Ramsey and strongly Ramsey cardinals},<br /> NOTE= {In preparation}}<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:WholenessAxiom,<br /> AUTHOR = {Corazza, Paul},<br /> TITLE = {The gap between ${\rm I}_3$ 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 (03E65)},<br /> MRNUMBER = {MR2028926 (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 /> #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 (03C62 03E35)},<br /> MRNUMBER = {611394 (82i:03063)},<br /> MRREVIEWER = {F. R. 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Hungar.},<br /> FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br /> VOLUME = {13},<br /> YEAR = {1962},<br /> PAGES = {223--226},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.60},<br /> MRNUMBER = {0141603 (25 \#5001)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br /> AUTHOR = {Erd&amp;#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 /> VOLUME = {9},<br /> YEAR = {1958},<br /> PAGES = {111--131},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.00},<br /> MRNUMBER = {0095124 (20 \#1630)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #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 /> url = {https://arxiv.org/abs/1607.04904v4}<br /> eprint = {arXiv:1607.04904},<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 /> doi = {10.1002/malq.19960420109},<br /> volume = {42},<br /> 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 /> journal = {Mathematical Logic Quarterly},<br /> doi = {10.1002/malq.19970430309},<br /> volume = {43},<br /> pages = {369--377},<br /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19970430309/abstract}<br /> }<br /> <br /> #Esser99:ConsistencyPositiveTheory bibtex=@article{Esser96:ConsistencyPositiveTheory,<br /> author = {Esser, Olivier},<br /> 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 /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.19990450110/abstract}<br /> }<br /> <br /> #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 /> volume = {65},<br /> number = {4},<br /> pages = {1911--1916},<br /> doi = {10.2307/2695086},<br /> url = {http://www.jstor.org/stable/2695086}<br /> }<br /> <br /> #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 /> volume = {19},<br /> pages = {97--100},<br /> url = {http://onlinelibrary.wiley.com/doi/10.1002/malq.200310009/abstract}<br /> }<br /> <br /> #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 /> #Foreman2010:Handbook bibtex=@book<br /> {Foreman2010:Handbook, <br /> 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 /> <br /> #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. 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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 /> #GitmanJohnstone:IndestructiblyRamsey bibtex=@ARTICLE{GitmanJohnstone:IndestructiblyRamsey,<br /> AUTHOR= {Gitman, Victoria and Johnstone, Thomas},<br /> TITLE= {Indestructibility for Ramsey and Ramsey-like cardinals},<br /> NOTE= {In preparation}}<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. 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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 (03E65)},<br /> MRNUMBER = {1816602 (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 /> #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 (03E35)},<br /> MRNUMBER = {2489293 (2010g:03083)},<br /> MRREVIEWER = {Carlos A. Di Prisco},<br /> DOI = {10.1002/malq.200710084},<br /> URL = {http://boolesrings.org/hamkins/tallcardinals/},<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 /> url={https://arxiv.org/pdf/math/9909029.pdf/},<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 /> 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 /> url = {https://arxiv.org/pdf/1403.2788.pdf},<br /> eprint = {1403.2788},<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(83)90020-9},<br /> url={https://ac.els-cdn.com/0168007283900209/1-s2.0-0168007283900209-main.pdf?_tid=466bc36a-c95e-11e7-bf33-00000aab0f27&amp;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 /> doi = {10.1016/0168-0072(89)90030-4},<br /> url={https://ac.els-cdn.com/0168007289900304/1-s2.0-0168007289900304-main.pdf?_tid=2f5a4ffe-e130-11e7-9794-00000aacb361&amp;acdnat=1513298453_24fe48742f365da91523f1174bb74117}<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 /> <br /> #JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson bibtex=@article{<br /> JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson,<br /> AUTHOR = {Jackson, Steve; Ketchersid, Richard; Schlutzenberg, Farmer; Woodin, W. 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Bell},<br /> url = {http://math.bu.edu/people/aki/e.pdf},<br /> }<br /> <br /> <br /> #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 /> YEAR = {2009},<br /> PAGES = {xxii+536},<br /> URL = {https://link.springer.com/book/10.1007%2F978-3-540-88867-3}<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'''(1978)}, <br /> year = {1978}, <br /> url = {http://math.bu.edu/people/aki/d.pdf},}<br /> <br /> #Kentaro2007:DoubleHelix bibtex=@article{Kentaro2007:DoubleHelix,<br /> AUTHOR = {Kentaro, Sato},<br /> TITLE = {Double helix in large large cardinals and iteration of<br /> elementary embeddings},<br /> SERIES = {Annals of Pure and Applied Logic},<br /> PUBLISHER = {Elsevier B.V.},<br /> YEAR = {2007},<br /> URL = {https://ac.els-cdn.com/S0168007207000127/1-s2.0-S0168007207000127-main.pdf?_tid=aa889390-c1e4-11e7-a507-00000aacb362&amp;acdnat=1509857531_549949bbb11277bb53825de297d7dc00},<br /> }<br /> <br /> #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 /> VOLUME = {43},<br /> NUMBER = {1},<br /> PAGES = {65--76},<br /> URL = {http://www.jstor.org/stable/2271949}<br /> }<br /> <br /> #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 /> editor = {Foreman, Mathew; Kanamori, Akihiro},<br /> year = {2010},<br /> publisher = {Springer},<br /> url = {http://logic.harvard.edu/koellner/LCFD.pdf}<br /> }<br /> <br /> #Larson2010:HistoryDeterminacy bibtex=@article{<br /> {Larson2010:HistoryDeterminacy,<br /> AUTHOR = {Larson, Paul B.},<br /> TITLE = {A brief history of determinacy},<br /> YEAR = {2013},<br /> URL = {http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf}<br /> }<br /> <br /> #Laver1997:Implications bibtex=@article {Laver1997:Implications,<br /> AUTHOR = {Laver, Richard},<br /> TITLE = {Implications between strong large cardinal axioms},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {90},<br /> YEAR = {1997},<br /> NUMBER = {1--3},<br /> PAGES = {79--90},<br /> ISSN = {0168-0072},<br /> MRCLASS = {03E55 (03E35)},<br /> MRNUMBER = {1489305 (99c:03074)},<br /> MRREVIEWER = {Douglas R. Burke},<br /> }<br /> <br /> #Maddy88:BelAxiomsI bibtex=@article{Maddy88:BelAxiomsI,<br /> AUTHOR = {Maddy, Penelope},<br /> TITLE = {Believing the axioms. I},<br /> JOURNAL = {J. Symbolic Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {2},<br /> PAGES = {181--511},<br /> DOI = {10.2307/2274520},<br /> URL = {http://www.jstor.org/stable/2274520}<br /> }<br /> #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 /> VOLUME = {53},<br /> YEAR = {1988},<br /> NUMBER = {3},<br /> PAGES = {736--764},<br /> DOI = {10.2307/2274569},<br /> URL = {http://www.jstor.org/stable/2274569}<br /> }<br /> #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 /> }<br /> <br /> #Makowsky1985:CompactLogics bibtex=@article{Makowsky1985:CompactLogics,<br /> AUTHOR = {Makowsky, Johann},<br /> TITLE = {Vopěnka's Principle and Compact Logics},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents<br /> YEAR = {1985},<br /> }<br /> <br /> #Mitchell1997:JonssonErdosCoreModel bibtex=@article{#Mitchell1997:JonssonErdosCoreModel,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {Jónsson Cardinals, Erdős Cardinals, and the Core Model},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = {https://arxiv.org/pdf/math/9706207.pdf},<br /> YEAR = {1997}<br /> }<br /> <br /> #Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {The Covering Lemma},<br /> JOURNAL = {Handbook of Set Theory},<br /> EDITOR = {M. Foreman and A. Kanamori and M. Magidor},<br /> URL = {http://www.math.cas.cz/~jech/library/mitchell/covering.ps},<br /> YEAR = {2001}<br /> }<br /> <br /> #Miyamoto1998:ANoteOnWeakSegmentsOfPFA bibtex=@article{Miyamoto1998:ANoteOnWeakSegmentsOfPFA,<br /> AUTHOR = {Miyamoto, Tadatoshi}.<br /> TITLE = {A note on weak segments of PFA},<br /> JOURNAL = {Proceedings of the sixth Asian logic conference},<br /> YEAR = {1998},<br /> PAGES = {175--197}<br /> }<br /> <br /> #Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge bibtex=@article{Perlmutter2010:TheLargeCardinalsBetweenSupercompactAlmostHuge,<br /> AUTHOR = {Perlmutter, Norman}.<br /> TITLE = {The large cardinals between supercompact and almost-huge},<br /> YEAR = {2010},<br /> URL = {https://arxiv.org/pdf/1307.7387.pdf},<br /> }<br /> <br /> <br /> #Rathjen2006:OrdinalAnalysis bibtex=@article{Rathjen2006:OrdinalAnalysis,<br /> AUTHOR = {Rathjen, Michael}.<br /> TITLE = {The art of ordinal analysis},<br /> YEAR = {2006},<br /> URL = {http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf}<br /> }<br /> <br /> #SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br /> AUTHOR = {Sharpe, Ian and Welch, Philip},<br /> TITLE = {Greatly Erd&amp;#337;s cardinals with some generalizations to<br /> the Chang and Ramsey properties},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {162},<br /> YEAR = {2011},<br /> NUMBER = {11},<br /> PAGES = {863--902},<br /> ISSN = {0168-0072},<br /> CODEN = {APALD7},<br /> MRCLASS = {03E04 (03E35 03E45 03E55)},<br /> MRNUMBER = {2817562},<br /> DOI = {10.1016/j.apal.2011.04.002},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2011.04.002},<br /> }<br /> <br /> #Shelah1994:CardinalArithmetic bibtex=@article {#Shelah1994:CardinalArithmetic,<br /> AUTHOR = {Shelah, Saharon},<br /> TITLE = {Cardinal Arithmetic},<br /> JOURNAL = {Oxford Logic Guides},<br /> VOLUME = {29},<br /> YEAR = {1994},<br /> }<br /> <br /> #Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Partial near supercompactness},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> JOURNAL = {Ann. Pure Appl. Logic},<br /> DOI = {10.1016/j.apal.2012.08.001},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2012.08.001},<br /> YEAR = {2012},<br /> NOTE = {In Press.}<br /> }<br /> <br /> #Schanker2011:WeaklyMeasurableCardinals bibtex=@article{Schanker2011:WeaklyMeasurableCardinals,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals},<br /> YEAR = {2011},<br /> JOURNAL = {MLQ Math. Log. Q.},<br /> FJOURNAL = {MLQ. Mathematical Logic Quarterly},<br /> VOLUME = {57},<br /> NUMBER = {3},<br /> PAGES = {266--280},<br /> DOI = {10.1002/malq.201010006},<br /> URL = {http://dx.doi.org/10.1002/malq.201010006}<br /> }<br /> <br /> #Schanker2011:Thesis bibtex=@phdthesis{Schanker2011:Thesis,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals and partial near supercompactness},<br /> SCHOOL = {CUNY Graduate Center},<br /> YEAR = {2011}<br /> }<br /> #Schindler2000:RemarkableCardinal bibtex=@article {Schindler2000:RemarkableCardinal,<br /> AUTHOR = {Schindler, Ralf-Dieter},<br /> TITLE = {Proper forcing and remarkable cardinals},<br /> JOURNAL = {Bull. Symbolic Logic},<br /> FJOURNAL = {The Bulletin of Symbolic Logic},<br /> VOLUME = {6},<br /> YEAR = {2000},<br /> NUMBER = {2},<br /> PAGES = {176--184},<br /> ISSN = {1079-8986},<br /> MRCLASS = {03E40 (03E45 03E55)},<br /> MRNUMBER = {1765054 (2001h:03096)},<br /> MRREVIEWER = {A. Kanamori},<br /> DOI = {10.2307/421205},<br /> URL = {http://dx.doi.org/10.2307/421205},<br /> }<br /> #Silver1970:ErdosCardinal bibtex=@article {MR0274278,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {A large cardinal in the constructible universe},<br /> JOURNAL = {Fund. Math.},<br /> FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br /> VOLUME = {69},<br /> YEAR = {1970},<br /> PAGES = {93--100},<br /> ISSN = {0016-2736},<br /> MRCLASS = {02.65},<br /> MRNUMBER = {0274278 (43 \#43)},<br /> MRREVIEWER = {N. C. A. da Costa},<br /> }<br /> #Silver1971:ZeroSharp bibtex=@article {MR0409188,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {Some applications of model theory in set theory},<br /> JOURNAL = {Ann. Math. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {3},<br /> YEAR = {1971},<br /> NUMBER = {1},<br /> PAGES = {45--110},<br /> ISSN = {0168-0072},<br /> MRCLASS = {02K35},<br /> MRNUMBER = {0409188 (53 \#12950)},<br /> MRREVIEWER = {Andreas Blass},<br /> }<br /> #Suzuki1998:NojVtoVinVofG bibtex=@article{Suzuki1998:NojVtoVinV[G],<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 (03E05)},<br /> MRNUMBER = {MR1650737 (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 (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 /> URL = {https://arxiv.org/abs/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 /> #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 (03E05 03E55)},<br /> MRNUMBER = {MR2838054 (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 /> URL = {https://arxiv.org/pdf/math/9611209.pdf},<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&amp;rep=rep1&amp;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 /> #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 /> <br /> URL = {http://www.worldscientific.com/doi/abs/10.1142/S021906131000095X},<br /> eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br /> }<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 (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 /> <br /> &lt;/biblio&gt;<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> Zetapology http://cantorsattic.info/index.php?title=Library&diff=2685 Library 2018-10-24T16:26:26Z <p>Zetapology: added drake's set theory book</p> <hr /> <div>{{DISPLAYTITLE: The Cantor's attic library}}<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 /> &lt;!-- <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 /> //--&gt;<br /> <br /> &lt;biblio force=true&gt;<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 (04A20)},<br /> MRNUMBER = {0460120 (57 \#116)},<br /> MRREVIEWER = {Thomas J. Jech}<br /> }<br /> <br /> #BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@unpublished {BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br /> AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosicky, Jirí},<br /> TITLE = &quot;Definable orthogonality classes in accessible categories are small&quot;,<br /> NOTE = &quot;submitted for publication&quot;,<br /> url = {http://arxiv.org/abs/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 /> url = {http://jdh.hamkins.org/superstrong-never-indestructible/}<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 (Colloq., Keszthely, 1973; dedicated to P. Erd&amp;#337;s on his 60th birthday), Vol. I},<br /> PAGES = {109--130. Colloq. Math. Soc. J&amp;#225;nos Bolyai, Vol. 10},<br /> PUBLISHER = {North-Holland},<br /> ADDRESS = {Amsterdam},<br /> YEAR = {1975},<br /> MRCLASS = {02K35 (04A20)},<br /> MRNUMBER = {0384553 (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 = &quot;Blass, Andreas&quot;,<br /> fjournal = &quot;Pacific Journal of Mathematics&quot;,<br /> journal = &quot;Pacific J. Math.&quot;,<br /> number = &quot;2&quot;,<br /> pages = &quot;335--346&quot;,<br /> publisher = &quot;Pacific Journal of Mathematics, A Non-profit Corporation&quot;,<br /> title = &quot;Exact functors and measurable cardinals.&quot;,<br /> url = &quot;https://projecteuclid.org:443/euclid.pjm/1102867389&quot;,<br /> volume = &quot;63&quot;,<br /> year = &quot;1976&quot;<br /> }<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 /> url = {https://arxiv.org/pdf/1708.07561.pdf},<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 /> #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 /> url = {https://arxiv.org/pdf/1305.5961.pdf},<br /> }<br /> <br /> #CodyGitman:EastonTheoremRamsey bibtex=@ARTICLE{CodyGitman:EastonTheoremRamsey,<br /> AUTHOR= {Cody, Brent and Gitman, Victoria},<br /> TITLE= {Easton's theorem for Ramsey and strongly Ramsey cardinals},<br /> NOTE= {In preparation}}<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:WholenessAxiom,<br /> AUTHOR = {Corazza, Paul},<br /> TITLE = {The gap between ${\rm I}_3$ 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 (03E65)},<br /> MRNUMBER = {MR2028926 (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 /> #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 (03C62 03E35)},<br /> MRNUMBER = {611394 (82i:03063)},<br /> MRREVIEWER = {F. R. 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Hungar.},<br /> FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br /> VOLUME = {13},<br /> YEAR = {1962},<br /> PAGES = {223--226},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.60},<br /> MRNUMBER = {0141603 (25 \#5001)},<br /> MRREVIEWER = {L. Gillman},<br /> }<br /> <br /> #ErdosHajnal1958:ErdosCardinals bibtex=@article {ErdosHajnal1958:ErdosCardinals,<br /> AUTHOR = {Erd&amp;#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 /> VOLUME = {9},<br /> YEAR = {1958},<br /> PAGES = {111--131},<br /> ISSN = {0001-5954},<br /> MRCLASS = {04.00},<br /> MRNUMBER = {0095124 (20 \#1630)},<br /> MRREVIEWER = {L. 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Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents<br /> YEAR = {1985},<br /> }<br /> <br /> #Mitchell1997:JonssonErdosCoreModel bibtex=@article{#Mitchell1997:JonssonErdosCoreModel,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {Jónsson Cardinals, Erdős Cardinals, and the Core Model},<br /> JOURNAL = {J. Symbol Logic},<br /> FJOURNAL = {The Journal of Symbolic Logic},<br /> URL = {https://arxiv.org/pdf/math/9706207.pdf},<br /> YEAR = {1997}<br /> }<br /> <br /> #Mitchell2001:TheCoveringLemma bibtex=@article{Mitchell2001:TheCoveringLemma,<br /> AUTHOR = {Mitchell, William J.},<br /> TITLE = {The Covering Lemma},<br /> JOURNAL = {Handbook of Set Theory},<br /> EDITOR = {M. Foreman and A. Kanamori and M. 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Pure Appl. Logic},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> VOLUME = {162},<br /> YEAR = {2011},<br /> NUMBER = {11},<br /> PAGES = {863--902},<br /> ISSN = {0168-0072},<br /> CODEN = {APALD7},<br /> MRCLASS = {03E04 (03E35 03E45 03E55)},<br /> MRNUMBER = {2817562},<br /> DOI = {10.1016/j.apal.2011.04.002},<br /> URL = {http://dx.doi.org/10.1016/j.apal.2011.04.002},<br /> }<br /> <br /> #Shelah1994:CardinalArithmetic bibtex=@article {#Shelah1994:CardinalArithmetic,<br /> AUTHOR = {Shelah, Saharon},<br /> TITLE = {Cardinal Arithmetic},<br /> JOURNAL = {Oxford Logic Guides},<br /> VOLUME = {29},<br /> YEAR = {1994},<br /> }<br /> <br /> #Schanker:PartialNearSupercompactness bibtex=@ARTICLE{Schanker:PartialNearSupercompactness,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Partial near supercompactness},<br /> FJOURNAL = {Annals of Pure and Applied Logic},<br /> JOURNAL = {Ann. Pure Appl. 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Mathematical Logic Quarterly},<br /> VOLUME = {57},<br /> NUMBER = {3},<br /> PAGES = {266--280},<br /> DOI = {10.1002/malq.201010006},<br /> URL = {http://dx.doi.org/10.1002/malq.201010006}<br /> }<br /> <br /> #Schanker2011:Thesis bibtex=@phdthesis{Schanker2011:Thesis,<br /> AUTHOR = {Schanker, Jason A.},<br /> TITLE = {Weakly measurable cardinals and partial near supercompactness},<br /> SCHOOL = {CUNY Graduate Center},<br /> YEAR = {2011}<br /> }<br /> #Schindler2000:RemarkableCardinal bibtex=@article {Schindler2000:RemarkableCardinal,<br /> AUTHOR = {Schindler, Ralf-Dieter},<br /> TITLE = {Proper forcing and remarkable cardinals},<br /> JOURNAL = {Bull. Symbolic Logic},<br /> FJOURNAL = {The Bulletin of Symbolic Logic},<br /> VOLUME = {6},<br /> YEAR = {2000},<br /> NUMBER = {2},<br /> PAGES = {176--184},<br /> ISSN = {1079-8986},<br /> MRCLASS = {03E40 (03E45 03E55)},<br /> MRNUMBER = {1765054 (2001h:03096)},<br /> MRREVIEWER = {A. Kanamori},<br /> DOI = {10.2307/421205},<br /> URL = {http://dx.doi.org/10.2307/421205},<br /> }<br /> #Silver1970:ErdosCardinal bibtex=@article {MR0274278,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {A large cardinal in the constructible universe},<br /> JOURNAL = {Fund. Math.},<br /> FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br /> VOLUME = {69},<br /> YEAR = {1970},<br /> PAGES = {93--100},<br /> ISSN = {0016-2736},<br /> MRCLASS = {02.65},<br /> MRNUMBER = {0274278 (43 \#43)},<br /> MRREVIEWER = {N. C. A. da Costa},<br /> }<br /> #Silver1971:ZeroSharp bibtex=@article {MR0409188,<br /> AUTHOR = {Silver, Jack},<br /> TITLE = {Some applications of model theory in set theory},<br /> JOURNAL = {Ann. Math. 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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 (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 /> URL = {https://arxiv.org/abs/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 /> #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 (03E05 03E55)},<br /> MRNUMBER = {MR2838054 (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 /> URL = {https://arxiv.org/pdf/math/9611209.pdf},<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&amp;rep=rep1&amp;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 /> #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 /> <br /> URL = {http://www.worldscientific.com/doi/abs/10.1142/S021906131000095X},<br /> eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br /> }<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 (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 /> <br /> &lt;/biblio&gt;<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> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2684 Measurable 2018-10-23T23:01:59Z <p>Zetapology: /* Category Theoretic Characterization */ got rid of bs</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). Let $j:V\rightarrow V$ be as described. Then any $\Delta_1$ formula $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is also $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> <br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subseteq\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> <br /> Finally, this gives much more general (and astonishing) characterizations of measurability in terms of nontrivial elementary embeddings of classes into themselves. The following are equivalent:<br /> #$\kappa$ is measurable.<br /> #There is some transitive class (or set) $M\models\text{ZFC}$ with $\mathcal{P}^2(\kappa)\subseteq M$ and some $j:M\prec_{\Delta_0}M$ with critical point $\kappa$. <br /> #For every transitive class (or set) $M\models\text{ZFC}$ with $\kappa\subseteq M$ there is a $j:M\prec_{\Delta_1}M$ with critical point $\kappa$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 3). Let $\kappa$ be measurable and let $j:V\prec_{\Delta_1}V$. Then, let $M\models\text{ZFC}$ be a transitive class with $\kappa\subseteq M$. Consider $j\upharpoonright M:M\rightarrow M$. Let $\varphi$ be a $\Delta_1$ formula. Then, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[x,y,z...]$ (because $\Delta_1$ formulae are upward and downward absolute for transitive classes). Since $j$ is $\Delta_1$-elementary, for any $x,y,z...\in M$:$$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]\Leftrightarrow\varphi[j\upharpoonright M(x),j\upharpoonright M(y),j\upharpoonright M(z)...]$$ $$\Leftrightarrow M\models\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j\upharpoonright M(x),j\upharpoonright M(y),j\upharpoonright M(z)...]$$<br /> Therefore $j\upharpoonright M:M\prec_{\Delta_1}$. Now note that $j$ has critical point $\kappa$ and so $j\upharpoonright M$ does too.<br /> <br /> (3 implies 2). Simply choose $M=V$ and note that any $j:V\prec_{\Delta_1}V$ with critical point $\kappa$ is already a $j:M\prec_{\Delta_0}M$ with critical point $\kappa$.<br /> <br /> (2 implies 1). If $j:M\prec_{\Delta_0}M$ is as described, then simply choose $U=\{X\subseteq\kappa:\kappa\in j(X)\}$ as usual. The proof that $U$ is a $\kappa$-complete nonprincipal ultrafilter is a little more difficult than usual:<br /> *If $X\in U$ and $X\subseteq Y\subseteq\kappa$, then $M\models (j(X)\subseteq j(Y)\subseteq j(\kappa))$ so $j(X)\subseteq j(Y)\subseteq j(\kappa)$ and therefore $\kappa\in j(Y)$ and $Y\in U$ ($U$ is closed upwards under subset).<br /> *If $X,Y\in U$ then $\kappa\in j(X)\cap j(Y)$ and by $\Delta_0$-ness of $\cap$, $j(X\cap Y)=j(X)\cap j(Y)$ so $\kappa\in j(X\cap Y)$ and therefore $X\cap Y\in U$ ($U$ is a filter).<br /> *Assume $X\in U$ is a finite set. Then it is easily verified that $j(X)=j&quot;X$ and so $\kappa\in j&quot;X$. This is a contradiction as $\kappa\neq j(x)$ for any set $x$; if it were then it would be an ordinal below $\kappa$ and so $\kappa$ would not be the critical point. ($U$ is nonprincipal).<br /> *If $X\notin U$ then $\kappa\notin j(X)$ so $\kappa\in (j(\kappa)\setminus j(X))$ and by $\Delta_0$-ness of $\setminus$, $\kappa\in j(\kappa\setminus X)$ meaning $\kappa\setminus X\in U$. ($U$ is an ultrafilter).<br /> *Finally, if $F\subseteq U$ is a family of size $\lambda&lt;\kappa$, then $j(F)=j&quot;F$ and since $\kappa\in j(X)$ for every $X\in F$, $\kappa\in X$ for every $X\in j&quot;F=j(F)$ and so $\kappa\in\cap j(F)$. By $\Delta_0$-ness of $\cap$, $\kappa\in j(\cap F)$ and so $\cap F\in U$. ($U$ is $\kappa$-complete).<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$ &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;.<br /> <br /> For more information, read &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;.<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$. <br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2683 Measurable 2018-10-22T19:50:36Z <p>Zetapology: /* Other Embedding Characterizations */ added another embedding char</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). Let $j:V\rightarrow V$ be as described. Then any $\Delta_1$ formula $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is also $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> <br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subseteq\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> <br /> Finally, this gives much more general (and astonishing) characterizations of measurability in terms of nontrivial elementary embeddings of classes into themselves. The following are equivalent:<br /> #$\kappa$ is measurable.<br /> #There is some transitive class (or set) $M\models\text{ZFC}$ with $\mathcal{P}^2(\kappa)\subseteq M$ and some $j:M\prec_{\Delta_0}M$ with critical point $\kappa$. <br /> #For every transitive class (or set) $M\models\text{ZFC}$ with $\kappa\subseteq M$ there is a $j:M\prec_{\Delta_1}M$ with critical point $\kappa$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 3). Let $\kappa$ be measurable and let $j:V\prec_{\Delta_1}V$. Then, let $M\models\text{ZFC}$ be a transitive class with $\kappa\subseteq M$. Consider $j\upharpoonright M:M\rightarrow M$. Let $\varphi$ be a $\Delta_1$ formula. Then, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[x,y,z...]$ (because $\Delta_1$ formulae are upward and downward absolute for transitive classes). Since $j$ is $\Delta_1$-elementary, for any $x,y,z...\in M$:$$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]\Leftrightarrow\varphi[j\upharpoonright M(x),j\upharpoonright M(y),j\upharpoonright M(z)...]$$ $$\Leftrightarrow M\models\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j\upharpoonright M(x),j\upharpoonright M(y),j\upharpoonright M(z)...]$$<br /> Therefore $j\upharpoonright M:M\prec_{\Delta_1}$. Now note that $j$ has critical point $\kappa$ and so $j\upharpoonright M$ does too.<br /> <br /> (3 implies 2). Simply choose $M=V$ and note that any $j:V\prec_{\Delta_1}V$ with critical point $\kappa$ is already a $j:M\prec_{\Delta_0}M$ with critical point $\kappa$.<br /> <br /> (2 implies 1). If $j:M\prec_{\Delta_0}M$ is as described, then simply choose $U=\{X\subseteq\kappa:\kappa\in j(X)\}$ as usual. The proof that $U$ is a $\kappa$-complete nonprincipal ultrafilter is a little more difficult than usual:<br /> *If $X\in U$ and $X\subseteq Y\subseteq\kappa$, then $M\models (j(X)\subseteq j(Y)\subseteq j(\kappa))$ so $j(X)\subseteq j(Y)\subseteq j(\kappa)$ and therefore $\kappa\in j(Y)$ and $Y\in U$ ($U$ is closed upwards under subset).<br /> *If $X,Y\in U$ then $\kappa\in j(X)\cap j(Y)$ and by $\Delta_0$-ness of $\cap$, $j(X\cap Y)=j(X)\cap j(Y)$ so $\kappa\in j(X\cap Y)$ and therefore $X\cap Y\in U$ ($U$ is a filter).<br /> *Assume $X\in U$ is a finite set. Then it is easily verified that $j(X)=j&quot;X$ and so $\kappa\in j&quot;X$. This is a contradiction as $\kappa\neq j(x)$ for any set $x$; if it were then it would be an ordinal below $\kappa$ and so $\kappa$ would not be the critical point. ($U$ is nonprincipal).<br /> *If $X\notin U$ then $\kappa\notin j(X)$ so $\kappa\in (j(\kappa)\setminus j(X))$ and by $\Delta_0$-ness of $\setminus$, $\kappa\in j(\kappa\setminus X)$ meaning $\kappa\setminus X\in U$. ($U$ is an ultrafilter).<br /> *Finally, if $F\subseteq U$ is a family of size $\lambda&lt;\kappa$, then $j(F)=j&quot;F$ and since $\kappa\in j(X)$ for every $X\in F$, $\kappa\in X$ for every $X\in j&quot;F=j(F)$ and so $\kappa\in\cap j(F)$. By $\Delta_0$-ness of $\cap$, $\kappa\in j(\cap F)$ and so $\cap F\in U$. ($U$ is $\kappa$-complete).<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$ &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> For more information, read &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;.<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$. <br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Cellar&diff=2682 Cellar 2018-10-20T16:37:16Z <p>Zetapology: /* Definition of infinity */ added first steps</p> <hr /> <div>{{DISPLAYTITLE: The cellar}}<br /> <br /> This page will contain links to summary accounts of supporting foundational or background material used on the rest of the site. <br /> <br /> <br /> You may like to begin in the [[playroom]] for an entertaining introduction to infinity. <br /> <br /> Meanwhile, we expect that this page and these resources will be expanded as Cantor's attic develops.<br /> <br /> == Definition of infinity ==<br /> <br /> * Short informal presentation of the concept of [[infinity]].<br /> * The [[first steps]] towards infinity.<br /> <br /> == Elementary set-theoretic topics == <br /> <br /> * [[transitive]]<br /> * [[Ordering Relations|Basic Order Theory]]<br /> * [[ordinal]]<br /> * [[successor ordinal]]<br /> * [[limit ordinal]]<br /> * [[cardinality]]<br /> * [[axiom of choice]]<br /> * [[stationary]], [[club]]<br /> * [[Hereditary cardinality]]<br /> * [[ultrafilter]], [[measure]]<br /> * [[ultrapower]]<br /> * [[Partition property]]<br /> <br /> == Axiomatic set theories ==<br /> <br /> * [[Morse-Kelley set theory]]<br /> * [[ZFC|Zermelo-Fraenkel set theory]]<br /> * [[Positive set theory]]<br /> * [[Kripke-Platek|Kripke-Platek set theory]]<br /> <br /> == Forcing == <br /> <br /> * [[forcing]]<br /> * [[Boolean-valued models]]<br /> * [[Boolean ultrapowers]]<br /> <br /> == Canonical inner models ==<br /> <br /> * The [[core model]]<br /> * The canonical model [[constructible universe | $L[\mu]$]] of one measurable cardinal <br /> * [[HOD]]<br /> * The [[constructible universe]]</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2681 Measurable 2018-10-20T16:27:51Z <p>Zetapology: added some sources</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). Let $j:V\rightarrow V$ be as described. Then any $\Delta_1$ formula $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is also $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> <br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subseteq\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$ &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> For more information, read &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;.<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$. <br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2680 Measurable 2018-10-20T16:24:34Z <p>Zetapology: /* Category Theoretic Characterization */ added source</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). Let $j:V\rightarrow V$ be as described. Then any $\Delta_1$ formula $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is also $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> <br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subseteq\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$ &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> For more information, read &lt;cite&gt;Blass1976:ExactFunctors&lt;/cite&gt;.<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Library&diff=2679 Library 2018-10-20T16:22:04Z <p>Zetapology: removed duplicate, added Exact Functors and Measurable Cardinals</p> <hr /> <div>{{DISPLAYTITLE: The Cantor's attic library}}<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 /> &lt;!-- <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 /> //--&gt;<br /> <br /> &lt;biblio force=true&gt;<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 (04A20)},<br /> MRNUMBER = {0460120 (57 \#116)},<br /> MRREVIEWER = {Thomas J. Jech}<br /> }<br /> <br /> #BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses bibtex=@unpublished {BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses,<br /> AUTHOR = {Bagaria, Joan and Casacuberta, Carles and Mathias, A. R. D. and Rosicky, Jirí},<br /> TITLE = &quot;Definable orthogonality classes in accessible categories are small&quot;,<br /> NOTE = &quot;submitted for publication&quot;,<br /> url = {http://arxiv.org/abs/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 /> url = {http://jdh.hamkins.org/superstrong-never-indestructible/}<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 (Colloq., Keszthely, 1973; dedicated to P. Erd&amp;#337;s on his 60th birthday), Vol. I},<br /> PAGES = {109--130. Colloq. Math. Soc. J&amp;#225;nos Bolyai, Vol. 10},<br /> PUBLISHER = {North-Holland},<br /> ADDRESS = {Amsterdam},<br /> YEAR = {1975},<br /> MRCLASS = {02K35 (04A20)},<br /> MRNUMBER = {0384553 (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 = &quot;Blass, Andreas&quot;,<br /> fjournal = &quot;Pacific Journal of Mathematics&quot;,<br /> journal = &quot;Pacific J. 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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 (03E65)},<br /> MRNUMBER = {MR2028926 (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 /> #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 (03C62 03E35)},<br /> MRNUMBER = {611394 (82i:03063)},<br /> MRREVIEWER = {F. R. 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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 /> #Foreman2010:Handbook bibtex=@book<br /> {Foreman2010:Handbook, <br /> 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 /> <br /> #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. 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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 (03E65)},<br /> MRNUMBER = {1816602 (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 /> #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 (03E35)},<br /> MRNUMBER = {2489293 (2010g:03083)},<br /> MRREVIEWER = {Carlos A. 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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 (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 /> URL = {https://arxiv.org/abs/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 /> #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 (03E05 03E55)},<br /> MRNUMBER = {MR2838054 (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 /> URL = {https://arxiv.org/pdf/math/9611209.pdf},<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&amp;rep=rep1&amp;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 /> #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 /> <br /> URL = {http://www.worldscientific.com/doi/abs/10.1142/S021906131000095X},<br /> eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021906131000095X}<br /> }<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 (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 /> <br /> &lt;/biblio&gt;<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> Zetapology http://cantorsattic.info/index.php?title=Ramsey&diff=2678 Ramsey 2018-10-20T16:07:19Z <p>Zetapology: /* Almost Ramsey cardinal */</p> <hr /> <div>{{DISPLAYTITLE: Ramsey cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Partition property]]<br /> Ramsey cardinals were introduced by Erd&amp;#337;s and Hajnal in &lt;cite&gt;ErdosHajnal1962:Ramsey&lt;/cite&gt;. A cardinal $\kappa$ is Ramsey if it has the partition property $\kappa\rightarrow (\kappa)^{\lt\omega}_2$.<br /> <br /> For infinite cardinals $\kappa$ and $\lambda$, the [[partition property]] $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is Ramsey, then $\kappa\rightarrow (\kappa)^{\lt\omega}_\lambda$ for every $\lambda&lt;\kappa$. Ramsey cardinals were named in honor of Frank Ramsey, whose Ramsey theorem for partition properties of $\omega$ motivated the generalizations of these to uncountable cardinals. A Ramsey cardinal $\kappa$ is exactly the [[Erdos|$\kappa$-Erd&amp;#337;s]] cardinal.<br /> <br /> Ramsey cardinals have a number of other characterizations. They may be characterized model theoretically through the existence of $\kappa$-sized sets of indiscernibles for models meeting the criteria discussed below, as well as through the existence of $\kappa$-sized models of set theory without power set with iterable ultrapowers.<br /> <br /> '''Indiscernibles''':<br /> Suppose $\mathcal A=(A,\ldots)$ is a model of a language $\mathcal L$ of size less than $\kappa$ whose universe $A$ contains $\kappa$ as a subset.<br /> <br /> If a cardinal $\kappa$ is Ramsey, then every such model $\mathcal A$ has a $\kappa$-sized set of indiscernibles $H\subseteq\kappa$, that is, for every formula $\varphi(\overline x)$ of $\mathcal L$ and every pair of tuples $\overline \alpha$ and $\overline \beta$ of elements of $H$, we have $\mathcal A\models\varphi (\overline \alpha)\leftrightarrow \varphi(\overline \beta)$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> '''Good sets of indiscernibles''':<br /> Suppose $A\subseteq\kappa$ and $L_\kappa[A]$ denotes the $\kappa^{\text{th}}$-level of the universe constructible using a predicate for $A$. A set $I\subseteq\kappa$ is a good set of indiscernibles for the model $\langle L_\kappa[A],A\rangle$ if for all $\gamma\in I$,<br /> * $\langle L_\gamma[A\cap \gamma],A\cap \gamma\rangle\prec \langle L_\kappa[A], A\rangle$,<br /> *$I\setminus\gamma$ is a set of indiscernibles for the model $\langle L_\kappa[A], A,\xi\rangle_{\xi\in\gamma}$.<br /> A cardinal $\kappa$ is Ramsey if and only if for every $A\subseteq\kappa$, there is a $\kappa$-sized good set of indiscernibles for the model $\langle L_\kappa[A], A\rangle$. &lt;cite&gt;DoddJensen1982:CoreModel&lt;/cite&gt;<br /> <br /> '''$M$-ultrafilters''': Suppose a transitive $M\models {\rm ZFC}^-$, the theory ${\rm ZFC}$ without the power set axiom (and using collection and separation rather than merely replacement) and $\kappa$ is a cardinal in $M$. We call $U\subseteq P(\kappa)^M$ an $M$-ultrafilter if the model $\langle M,U\rangle\models$&amp;#8220;$U$ is a normal ultrafilter on $\kappa$&amp;#8221;. In the case when the $M$-ultrafilter is not an element of $M$, the model $\langle M,U\rangle$ of $M$ together with a predicate for $U$ often fails to satisfy much of ${\rm ZFC}$. An $M$-ultrafilter $U$ is said to be weakly amenable (to $M$) if for every $A\in M$ of size $\kappa$ in $M$, the intersection $A\cap U$ is an element of $M$. An $M$-ultrafilter $U$ is countably complete if every countable sequence (possibly external to $M$) of elements of $U$ has a non-empty intersection (even if the intersection is not itself an element of $M$). A weak $\kappa$-model is a transitive set $M\models {\rm ZFC}^-$ of size $\kappa$ and containing $\kappa$ as an element. A modified ultrapower construction using only functions on $\kappa$ that are elements of $M$ can be carried out with an $M$-ultrafilter. If the $M$-ultrafilter happens to be countably complete, then the standard argument shows that the ultrapower is well-founded. If the $M$-ultrafilter is moreover weakly amenable, then a weakly amenable ultrafilter on the image of $\kappa$ in the well-founded ultrapower can be constructed from images of the pieces of $U$ that are in $M$. The ultrapower construction may be iterated in this manner, taking direct limits at limit stages, and in this case the countable completeness of the $M$-ultrafilter ensures that every stage of the iteration produces a well-founded model. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; (Ch. 19)<br /> <br /> A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists a weakly amenable countably complete $M$-ultrafilter on $\kappa$. &lt;cite&gt;DoddJensen1982:CoreModel&lt;/cite&gt;<br /> <br /> ==Ramsey cardinals and the constructible universe==<br /> <br /> Ramsey cardinals imply that [[zero sharp | $0^\sharp$]] exists and hence there cannot be Ramsey cardinals in $L$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> ==Relations with other large cardinals==<br /> <br /> * [[Measurable]] cardinals are Ramsey and stationary limits of Ramsey cardinals. &lt;cite&gt;ErdosHajnal1962:Ramsey&lt;/cite&gt;<br /> * Ramsey cardinals are [[unfoldable]] (using the $M$-ultrafilters characterization) and stationary limits of unfoldable cardinals (as they are stationary limits of $\omega_1$-iterable cardinals).<br /> * Ramsey cardinals are stationary limits of [[completely ineffable]] cardinals, they are [[Weakly ineffable |weakly ineffable]] but but the least Ramsey cardinal is not ineffable. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> <br /> ==Ramsey cardinals and forcing==<br /> <br /> *Ramsey cardinals are preserved by small forcing. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> *Ramsey cardinals $\kappa$ are preserved by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. &lt;cite&gt;GitmanJohnstone:IndestructiblyRamsey&lt;/cite&gt;<br /> *If $\kappa$ is Ramsey, there is a forcing extension in which $\kappa$ remains Ramsey and $2^\kappa\gt\kappa$. Indeed, if the ${\rm GCH}$ holds and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and satisfying $F(\alpha)\leq F(\beta)$ for $\alpha&lt;\beta$ and $\text{cf}(F(\alpha))&gt;\alpha$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey and $2^\delta=F(\delta)$ for every regular cardinal $\delta$. &lt;cite&gt;CodyGitman:EastonTheoremRamsey&lt;/cite&gt; <br /> * If the existence of Ramsey cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not Ramsey, but becomes Ramsey in a forcing extension. &lt;cite&gt;GitmanJohnstone:IndestructiblyRamsey&lt;/cite&gt;<br /> ==Ramsey-like cardinals==<br /> <br /> There are many Ramsey-like cardinals, most of which can be found in &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;.<br /> <br /> ===Super Ramsey cardinal===<br /> Super Ramsey cardinals were introduced by Gitman in &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;. They strengthen one definition of strong Ramseyness.<br /> <br /> A weak $\kappa$-model $M$ is a $\kappa$-model if additionally $M^{\lt\kappa}\subseteq M$. <br /> <br /> A cardinal $\kappa$ is '''super Ramsey''' if and only if for every $A\subseteq\kappa$, there is some $\kappa$-model $M$ with $A\subseteq M\prec H_{\kappa^+}$ such that there is some $N$ and some $\kappa$-powerset preserving nontrivial elementary embedding $j:M\prec N$.<br /> <br /> The following are some facts about super Ramsey cardinals:<br /> <br /> *[[Measurable]] cardinals are super Ramsey limits of super Ramsey cardinals. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> *Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey cardinals. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> *Super Ramseyness is downward absolute to $K$. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> *The required $M$ for a super Ramsey embedding is stationarily correct. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> *The consistency strength of a super Ramsey cardinal is stronger than that of a strongly Ramsey cardinal and weaker than that of a [[measurable]] cardinal. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> <br /> ===Strongly Ramsey cardinal===<br /> <br /> Strongly Ramsey cardinals were introduced by Gitman in &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;. They strengthen the $M$-ultrafilters characterization of Ramsey cardinals from weak $\kappa$-models to $\kappa$-models.<br /> <br /> A cardinal $\kappa$ is '''strongly Ramsey''' if every $A\subseteq\kappa$ is contained in a $\kappa$-model $M$ for which there exists a weakly amenable $M$-ultrafilter on $\kappa$. An $M$-ultrafilter for a $\kappa$-model $M$ is automatically countably complete since $\langle M,U\rangle$ satisfies that it is $\kappa$-complete and it must be correct about this since $M$ is closed under sequences of length less than $\kappa$.<br /> <br /> * [[Measurable]] cardinals are strongly Ramsey limits of strongly Ramsey cardinals. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> * Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey cardinals. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> * The least strongly Ramsey cardinal is not [[ineffable]]. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> * Forcing related properties of strongly Ramsey cardinals are the same as those of Ramsey cardinals described above. &lt;cite&gt;GitmanJohnstone:IndestructiblyRamsey&lt;/cite&gt;<br /> * The consistency strength of strongly Ramsey cardinals is stronger than that of Ramsey cardinals. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> * Strong Ramseyness is downward absolute to $K$. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> <br /> ===Virtually Ramsey cardinal===<br /> <br /> Virtually Ramsey cardinals were introduced by Sharpe and Welch in &lt;cite&gt;SharpeWelch2011:GreatlyErdosChang&lt;/cite&gt;. They weaken the good indiscernibles characterization of Ramsey cardinals and were motivated by finding an upper bound on the consistency strength of a variant of Chang's Conjecture studied in &lt;cite&gt;SharpeWelch2011:GreatlyErdosChang&lt;/cite&gt;. For $A\subseteq\kappa$, define that $\mathscr I=\{\alpha&lt;\kappa\mid$ there is an unbounded good set of indiscernibles $I_\alpha\subseteq\alpha$ for $\langle L_\kappa[A],A\rangle\}$. A cardinal $\kappa$ is virtually Ramsey if for every $A\subseteq\kappa$, the set $\mathscr I$ contains a club of $\kappa$.<br /> <br /> Virtually Ramsey cardinals are [[Mahlo]] and a virtually Ramsey cardinal that is [[weakly compact]] is already Ramsey. It is consistent from a Ramsey cardinal that there is a virtually Ramsey cardinal that is not Ramsey. It is open whether virtually Ramsey cardinals are weaker than Ramsey cardinals. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> <br /> ===Almost Ramsey cardinal===<br /> <br /> An uncountable cardinal $\kappa$ is '''almost Ramsey''' if and only if $\kappa\rightarrow(\alpha)^{&lt;\omega}$ for every $\alpha&lt;\kappa$. Equivalently:<br /> <br /> *$\kappa\rightarrow(\alpha)^{&lt;\omega}_\lambda$ for every $\alpha,\lambda&lt;\kappa$ <br /> *For every structure $\mathcal{M}$ with language of size $&lt;\kappa$, there is are sets of indiscernibles $I\subseteq\kappa$ for $\mathcal{M}$ of any size $&lt;\kappa$.<br /> *For every $\alpha&lt;\kappa$, $\eta_\alpha$ exists and $\eta_\alpha&lt;\kappa$.<br /> *$\kappa=\text{sup}\{\eta_\alpha:\alpha&lt;\kappa\}$ <br /> <br /> Where $\eta_\alpha$ is the [[Erdos|$\alpha$-Erd&amp;#337;s]] cardinal.<br /> <br /> Every almost Ramsey cardinal is a [[Beth|$\beth$-fixed point]], but the least almost Ramsey cardinal, if it exists, has cofinality $\omega$. In fact, the least almost Ramsey cardinal is not [[weakly inaccessible]], [[worldly]], or [[correct]]. However, if the least almost Ramsey cardinal exists, it is larger than the least [[Erdos|$\omega_1$-Erd&amp;#337;s]] cardinal. Any regular almost Ramsey cardinal is worldly, and any worldly almost Ramsey cardinal $\kappa$ has $\kappa$ almost Ramsey cardinals below it. <br /> <br /> The existence of a worldly almost Ramsey cardinal is stronger than the existence of a proper class of almost Ramsey cardinals. Therefore, the existence of a Ramsey cardinal is stronger than the existence of a proper class of almost Ramsey cardinals. The existence of a proper class of almost Ramsey cardinals is equivalent to the existence of $\eta_\alpha$ for every $\alpha$. The existence of an almost Ramsey cardinal is stronger than the existence of an $\omega_1$-Erd&amp;#337;s cardinal.<br /> <br /> The existence of an almost Ramsey cardinal is equivalent to the existence of $\eta^n(\omega)$ for every $n&lt;\omega$. On one hand, if a almost Ramsey cardinal $\kappa$ exists, then $\omega&lt;\kappa$. Then, $\eta_\omega$ is less than $\kappa$. Then, $\eta_{\eta_\omega}$ exists and is less than $\kappa$, and so on. On the other hand, if $\eta^n(\omega)$ exists for every $n&lt;\omega$, then $\text{sup}\{\eta^n(\omega):n&lt;\omega\}$ is almost Ramsey, and in fact the least almost Ramsey cardinal. Note that such a set exists by replacement and has a supremum by union.<br /> <br /> The Ramsey cardinals are precisely the [[Erdos|Erd&amp;#337;s]] almost Ramsey cardinals and also precisely the [[weakly compact]] almost Ramsey cardinals.<br /> <br /> ===$\alpha$-iterable cardinal===<br /> <br /> The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were introduced by Gitman in &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;. They form a hierarchy of large cardinal notions strengthening [[weakly compact]] cardinals, while weakening the $M$-ultrafilters characterization of Ramsey cardinals. Recall that if $\kappa$ is Ramsey, then every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $M$-ultrafilter, the ultrapower construction with which may be iterated through all the ordinals. Suppose $M$ is a weak $\kappa$-model and $U$ is an $M$-ultrafilter on $\kappa$. Define that:<br /> * $U$ is $0$-good if the ultrapower is well-founded,<br /> * $U$ is 1-good if it is 0-good and weakly amenable,<br /> * for an ordinal $\alpha&gt;1$, $U$ is $\alpha$-good, if it produces at least $\alpha$-many well-founded iterated ultrapowers. <br /> Using a theorem of Gaifman &lt;cite&gt;Gaifman1974:ElementaryEmbeddings&lt;/cite&gt;, if an $M$-ultrafilter is $\omega_1$-good, then it is already $\alpha$-good for every ordinal $\alpha$. <br /> <br /> For $1\leq\alpha\leq\omega_1$, a cardinal $\kappa$ is '''$\alpha$-iterable''' if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $\alpha$-good $M$-ultrafilter on $\kappa$. <br /> The $\alpha$-iterable cardinals form a hierarchy of strength above weakly compact cardinals and below Ramsey cardinals.<br /> <br /> The $1$-iterable cardinals are sometimes called the '''weakly Ramsey''' cardinals.<br /> <br /> * $1$-iterable cardinals are [[weakly ineffable]] and stationary limits of [[completely ineffable]] cardinals. The least $1$-iterable cardinal is not ineffable. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt;<br /> * An $\alpha$-iterable cardinal is $\beta$-iterable and a stationary limit of $\beta$-iterable cardinals for every $\beta&lt;\alpha$. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> * A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of $\omega_1$-iterable cardinals. This is already true of an $\omega_1$-[[Erdos | Erd&amp;#337;s]] cardinal. &lt;cite&gt;SharpeWelch2011:GreatlyErdosChang&lt;/cite&gt;<br /> * It is consistent from an $\omega$-[[Erdos | Erd&amp;#337;s]] cardinal that for every $n\in\omega$, there is a proper class of $n$-iterable cardinals.<br /> * A $2$-iterable cardinal is a limit of [[remarkable]] cardinals. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> * A [[remarkable]] cardinal implies the consistency of a $1$-iterable cardinal. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> * $\omega_1$-iterable cardinals imply that [[zero sharp | $0^\sharp$]] exists and hence there cannot be $\omega_1$-iterable cardinals in $L$. For $L$-countable $\alpha$, the $\alpha$-iterable cardinals are downward absolute to $L$. In fact, if [[zero sharp | $0^\sharp$]] exists, then every Silver indiscernible is $\alpha$-iterable in $L$ for every $L$-countable $\alpha$. &lt;cite&gt;GitmanWelch2011:RamseyLikeCardinalsII&lt;/cite&gt;<br /> * $\alpha$-iterable cardinals $\kappa$ are preserved by small forcing, by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. If $\kappa$ is $\alpha$-iterable, there is a forcing extension in which $\kappa$ remains $\alpha$-iterable and $2^\kappa\gt\kappa$. &lt;cite&gt;GitmanJohnstone:IndestructibleRamsey&lt;/cite&gt;<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2677 Measurable 2018-10-20T16:04:02Z <p>Zetapology: /* Other Embedding Characterizations */ syntax/making it sound better</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). Let $j:V\rightarrow V$ be as described. Then any $\Delta_1$ formula $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is also $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> <br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subseteq\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2676 Measurable 2018-10-20T15:47:43Z <p>Zetapology: /* Other Embedding Characterizations */ grammar</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). If $j:V\rightarrow V$ is as described then for any $\Delta_1$ formula, $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subset\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2675 Measurable 2018-10-20T15:45:27Z <p>Zetapology: /* Definitions */ added category theoretic characterization</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). If $j:V\rightarrow V$ is as described then for any $\Delta_1$ formula, $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subset\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff there it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> === Category Theoretic Characterization ===<br /> <br /> Interestingly, measurable cardinals have certain category theoretic properties about them. This connection is likely at heart due to the fact that certain embeddings $j:V\rightarrow V$ share connections with functors $F:\text{Set}\rightarrow\text{Set}$, and measurable cardinals can be characterized in terms of those embeddings.<br /> <br /> Specifically a measurable cardinal exists if and only if there is a nonidentity exact functor $F:\text{Set}\rightarrow\text{Set}$. In fact, although it is not directly stated in the paper, it is implied that '''a cardinal $\kappa$ is measurable if and only if there is an exact functor $F:\text{Set}\rightarrow\text{Set}$ such that $\kappa$ is the least cardinal for which $F$ does not preserve $\kappa$-indexed coproducts.'''<br /> <br /> ''How it was proven:''<br /> Assume $\kappa$ is measurable. Then, let $U$ be the $\kappa$-complete nonprincipal ultrafilter on $\kappa$. The reduced power $U$-prod then has many nice properties:<br /> #$U$-prod is an exact endofunctor on $\text{Set}$. (It preserves equalizers if and only if $U$ is $\sigma$-complete.)<br /> #$U$-prod preserves $\lambda$-indexed coproducts for any $\lambda&lt;\kappa$.<br /> #Assuming $U$-prod preserves $\kappa$-indexed coproducts, then it is shown that $U$ is $\kappa^+$-complete (which is impossible without making the ultrafilter improper). Therefore $U$-prod does not preserve $\kappa$-indexed coproducts.<br /> <br /> On the other hand, if $F$ is an exact endofunctor on $\text{Set}$, and $\kappa$ is the smallest cardinal for which $F$ does not preserve $\kappa$-indexed coproducts, then necessarily $F$ is shown to be naturally isomorphic to $U$-prod for some nonprincipal $\sigma$-complete ultrafilter on $\kappa$. Since for any $\lambda&lt;\kappa$, $F$ preserves $\lambda$-indexed coproducts, $U$-prod preserves $\lambda$-indexed coproducts, and $U$ is shown be $\lambda^+$-complete. Therefore $U$ is $\lambda^+$-complete for any $\lambda&lt;\kappa$ and so $U$ is $\kappa$-complete (and as shown, a nonprincipal ultrafilter).<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2674 Measurable 2018-10-20T02:36:52Z <p>Zetapology: /* Other Embedding Characterizations */</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Second-order|NBG]] or ZFC + $j$) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 4). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. Then, let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$-formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$. Therefore:<br /> $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (4 implies 3). If $j:V\rightarrow V$ is as described then for any $\Delta_1$ formula, $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subset\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). Namely, $\kappa$ is measurable iff there it is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> <br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Measurable&diff=2673 Measurable 2018-10-19T22:45:10Z <p>Zetapology: /* Definitions */</p> <hr /> <div>{{DISPLAYTITLE: Measurable cardinal}}<br /> A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to &quot;measure&quot; the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br /> <br /> Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br /> <br /> Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br /> <br /> == Definitions ==<br /> <br /> There are essentially two ways to &quot;measure&quot; a cardinal $\kappa$, that's to say we can require the measure to be $\sigma$-additive (a &quot;classical&quot; measure) or to be $\kappa$-additive (for every cardinal $\lambda$ such that $\lambda &lt; \kappa$, the union of $\lambda$ null sets still has measure zero).<br /> <br /> Let $\kappa$ be an uncountable cardinal.<br /> <br /> Theorem 1 : The following are equivalent :<br /> # There exists a 2-valued ($\sigma$-additive) measure on $\kappa$.<br /> # There exists a $\sigma$-complete nonprincipal ultrafilter on $\kappa$.<br /> <br /> The equivalence is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br /> <br /> An uncountable cardinal which satisfies the equivalent conditions of theorem 1 is sometimes called a 2-measurable cardinal (because &quot;2-valued&quot;). This is not a traditional notation, but it was used in an article of Gustave Choquet : &quot;Cardinaux 2-mesurables et cônes faiblement compacts&quot;, Annales de l'Institut Fourier, tome 17, n°2 (1967), P.383-393.<br /> <br /> Note : It is clear that, if $\kappa$ is 2-measurable, then every cardinal $\lambda$ such that $\lambda &gt; \kappa$ is also 2-measurable. <br /> Thus, the notion of 2-measurability separates the class $C$ of all cardinals in two subclasses : the &quot;moderated&quot; cardinals and the 2-measurable cardinals, the first one being an initial segment of $C$, and therefore this notion is of weak interest for the study of the hierarchy of large cardinals.<br /> <br /> === Embedding Characterization ===<br /> <br /> Theorem 2 : The following are equivalent :<br /> # There exists a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.<br /> # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br /> # There exists a nonprincipal ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded.<br /> <br /> To see that the second condition implies the first one, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br /> <br /> An uncountable cardinal $\kappa$ is called measurable if the equivalent conditions of theorem 2 are satisfied.<br /> <br /> The two theorems are related by the fact (easy to prove) that the least cardinal $\kappa$ (if it exists) which carries a $\sigma$-complete nonprincipal ultrafilter is measurable, and in this case every $\sigma$-complete nonprincipal ultrafilter on $\kappa$ is $\kappa$-complete (see for example Patrick Dehornoy : &quot;La théorie des ensembles&quot;, Calvage et Mounet, 2017).<br /> <br /> In other words, the first 2-measurable cardinal is measurable.<br /> <br /> Therefore, the two notions are equiconsistent, but in the general case they differ : every measurable cardinal is 2-measurable, and the converse is false.<br /> <br /> === Hayut Property ===<br /> <br /> There is also another quite interesting model-theoretic characterization of measurability. Let a theory $T$ be $\kappa$-unboundedly satisfiable iff for every $\lambda&lt;\kappa$, there is a model $\mathcal{M}\models T$ with $\lambda\leq|M|&lt;\kappa$. In other words, the sizes of models of $T$ are unbounded in $\kappa$.<br /> <br /> A class of formulae $Q$ is $\kappa$-Hayut iff for any $\kappa$-unboundedly satisfiable theory $T\subseteq Q$, there is a model of $T$ of size at least $\kappa$. More intuitively, $\kappa$-many small models of size less than $\kappa$ can combine to make one big $\kappa$-sized model.<br /> <br /> An abstract logic $\mathcal{L}$ is called almost $\kappa$-favorable iff there is some way to represent every sentence of $\mathcal{L}$ with vocabulary $\tau$ as a sequence of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ in such a way that the satisfaction relation is upward absolute for inner models $M$ of ZFC elementarily equivalent to $V$ with $M^{&lt;\kappa}\subset M$. If $\kappa$ is an uncountable regular cardinal, the following are almost $\kappa$-favorable:<br /> #$\mathcal{L}_{\lambda,\mu}$ for any $\lambda,\mu\leq\kappa$<br /> #$\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$, which is $\mathcal{L}_{\kappa,\omega}$ with universal cardinality quantifiers $q_\lambda$ for every $\lambda&lt;\kappa$ (where $M\models q_\lambda$ iff $|M|\geq q_\lambda$)<br /> #$\mathcal{L}_{\kappa,\kappa}$ with the addition of a single existential 2nd-order quantifier, where negation on the resulting sentences is not allowed<br /> <br /> Assuming $V=L$, every $\mathcal{L}$ where sentences are represented as sequences of length below $\kappa$ of symbols of $\tau$ and ordinals in $\kappa$ ($\kappa$-sequential logic) that has an extension with an $\mathcal{L}_{\omega,\omega}$-definable satisfaction relation is almost $\kappa$-favorable. For example: if $V=L$, then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is almost $\kappa$-favorable, but if a measurable exists then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not almost $\kappa$-favorable, and in fact if $\kappa$ is the least measurable then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is not $\kappa$-Hayut; however, if $\kappa$ is [[extendible]], then $\mathcal{L}_{\kappa,\kappa}^{&lt;\omega}$ is $\kappa$-Hayut, though it still isn't almost $\kappa$-favorable.<br /> <br /> An uncountable regular cardinal $\kappa$ is measurable if and only if $\mathcal{L}_{\kappa,\kappa}$ is $\kappa$-Hayut, if and only if $\mathcal{L}_{\kappa,\omega}(q_{&lt;\kappa})$ is $\kappa$-Hayut up to $2^\kappa$. Furthermore, an uncountable regular cardinal $\kappa$ is measurable if and only if every almost $\kappa$-favorable logic is $\kappa$-Hayut.<br /> <br /> For more information, see [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 this post].<br /> <br /> === Other Embedding Characterizations ===<br /> <br /> There are also other embedding characterizations of measurable cardinals. Namely (under [[Morse-Kelley set theory|MK]]) the following are equivalent for any cardinal $\kappa$:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0}V$.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_1}V$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow V$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$.<br /> #$\kappa$ is the critical point of some $j:V\rightarrow M$ for some inner model $M$ such that for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$. <br /> <br /> ''Proof:''<br /> <br /> (1 implies 5). If $\kappa$ is measurable, then $\kappa$ is the critical point of a $j:V\prec_M$ for some inner model $M$. Therefore $\kappa$ is the critical point of a $j:V\prec_{\Sigma_1}M$ and so for any $\Sigma_1$-formula $\varphi$, $\varphi[x,y,z...]\rightarrow M\models\varphi[j(x),j(y),j(z)...]$.<br /> <br /> (5 implies 4). If $j:V\rightarrow M$ is as described then let $\varphi$ be a $\Sigma_1$-formula. If $\varphi[x,y,z...]$ then<br /> $M\models\varphi[j(x),j(y),j(z)...]$ and because $\Sigma_1$ formulae are upward absolute for inner models, $\varphi[j(x),j(y),j(z)...]$.<br /> <br /> (4 implies 3). If $j:V\rightarrow V$ is as described then for any $\Delta_1$ formula, $\varphi$ is $\Sigma_1$ and $\neg\varphi$ is $\Sigma_1$. So: $$\varphi[x,y,z...]\rightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$\neg\varphi[x,y,z...]\rightarrow\neg\varphi[j(x),j(y),j(z)...]$$<br /> $$\therefore\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> <br /> (3 implies 2). If $j:V\prec_{\Delta_1}V$ then $j:V\prec_{\Delta_0}V$ by definition.<br /> (2 implies 1). If $j:V\prec_{\Delta_0}V$ has critical point $\kappa$, then $\{X\subset\kappa:\kappa\in j(X)\}$ is a $\kappa$-complete measure on $\kappa$. <br /> <br /> This gives further characterizations (which are weakenings of the $j:V\prec M$ characterizations). The following are equivalent:<br /> #$\kappa$ is measurable.<br /> #$\kappa$ is the critical point of some $j:V\prec_{\Delta_0} M$ for some inner model $M$.<br /> <br /> ''Proof:''<br /> <br /> (1 implies 2). Let $\kappa$ be measurable. Then $\kappa$ is the critical point of $j:V\prec M$ for some inner model $M$, and so $j:V\prec_{\Delta_0} M$.<br /> (2 implies 1). Let $\kappa$ be the critical point of a $j:V\prec_{\Delta_0}M$. Then, $j:V\rightarrow V$ and for any $\Delta_0$ formula $\varphi$, $\varphi[x,y,z...]\Leftrightarrow M\models\varphi[j(x),j(y),j(z)...]$. Of course, since $\varphi$ is $\Delta_0$, $M\models\varphi[j(x),j(y),j(z)...]$ iff $\varphi[j(x),j(y),j(z)...]$. So:<br /> $$\varphi[x,y,z...]\Leftrightarrow\varphi[j(x),j(y),j(z)...]$$<br /> $$j:V\prec_{\Delta_0}V$$<br /> $$\therefore\kappa\text{ is measurable.}$$<br /> <br /> == Properties ==<br /> <br /> If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br /> <br /> Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br /> <br /> If $\kappa$ is measurable and $\lambda&lt;\kappa$ then it cannot be true that $\kappa&lt;2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br /> <br /> If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br /> <br /> Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha&lt;\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br /> <br /> Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br /> <br /> If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br /> <br /> If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br /> <br /> ''See also: [[Ultrapower]]''<br /> <br /> === Failure of $\text{GCH}$ at a measurable ===<br /> <br /> Gitik proved that the following statements are equiconsistent:<br /> * The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa &gt; \kappa^+$<br /> * The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa &gt; \kappa^+$<br /> * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br /> <br /> Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br /> <br /> However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br /> <br /> == Real-valued measurable cardinal ==<br /> <br /> A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br /> <br /> If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br /> <br /> Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is [[forcing|generic extension]] in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br /> <br /> == See also ==<br /> * [[Ultrapower]]<br /> * [[Mitchell order]]<br /> * [[Axiom of determinacy]]<br /> * [[Strongly compact]] cardinal<br /> <br /> == Read more ==<br /> * Jech, Thomas - ''Set theory''<br /> <br /> * Bering A., Edgar - ''A brief introduction to measurable cardinals''</div> Zetapology http://cantorsattic.info/index.php?title=Constructible_universe&diff=2671 Constructible universe 2018-10-19T18:30:37Z <p>Zetapology: /* EM blueprints and alternative characterizations of $0^\#$ */</p> <hr /> <div>[[Category:Constructibility]]<br /> The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of [[ZFC|$\text{ZFC+}$]][[GCH|$\text{GCH}$]] (assuming the consistency of $\text{ZFC}$) showing that $\text{ZFC}$ cannot disprove $\text{GCH}$. It was then shown to be an important model of $\text{ZFC}$ for its satisfying of other axioms, thus making them consistent with $\text{ZFC}$. The idea is that $L$ is built up by ranks like $V$. $L_0$ is the empty set, and $L_{\alpha+1}$ is the set of all easily definable subsets of $L_\alpha$. The assumption that $V=L$ (also known as the '''Axiom of constructibility''') is undecidable from $\text{ZFC}$, and implies many axioms which are consistent with $\text{ZFC}$. A set $X$ is '''constructible''' iff $X\in L$. $V=L$ iff every set is constructible.<br /> <br /> == Definition ==<br /> <br /> $\mathrm{def}(X)$ is the set of all &quot;easily definable&quot; subsets of $X$ (specifically the $\Delta_0$ definable subsets). More specifically, a subset $x$ of $X$ is in $\mathrm{def}(X)$ iff there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,v_0,v_1...v_n]\}$. Then, $L_\alpha$ and $L$ are defined as follows:<br /> <br /> *$L_0=\emptyset$<br /> *$L_{\alpha+1}=\mathrm{def}(L_\alpha)$<br /> *$L_\beta=\bigcup_{\alpha&lt;\beta} L_\alpha$ if $\beta$ is a limit ordinal<br /> *$L=\bigcup_{\alpha\in\mathrm{Ord}} L_\alpha$<br /> <br /> === The Relativized constructible universes $L_\alpha(W)$ and $L_\alpha[W]$ ===<br /> <br /> $L_\alpha(W)$ for a class $W$ is defined the same way except $L_0(W)=\text{TC}(\{W\})$ (the transitive closure of $\{W\}$). $L_\alpha[W]$ for a class $W$ is defined in the same way as $L$ except using $\mathrm{def}_W(X)$, where $\mathrm{def}_W(X)$ is the set of all $x\subseteq X$ such that there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,W,v_0,v_1...v_n]\}$ (because the relativization of $\varphi$ to $X$ is used and $\langle X,\in\rangle$ is not used, this definition makes sense even when $W$ is not in $X$).<br /> <br /> $L[W]=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha[W]$ is always a model of $\text{ZFC}$, and always satisfies $\text{GCH}$ past a certain cardinality. $L(W)=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha(W)$ is always a model of $\text{ZF}$ but need not satisfy $\text{AC}$ (the axiom of choice). In particular, $L(\mathbb{R})$ is, under large cardinal assumptions, a model of the [[axiom of determinacy]]. However, Shelah proved that if $\lambda$ is a strong limit cardinal of uncountable cofinality then $L(\mathcal{P}(\lambda))$ is a model of $\text{AC}$.<br /> <br /> == The difference between $L_\alpha$ and $V_\alpha$ ==<br /> <br /> For $\alpha\leq\omega$, $L_\alpha=V_\alpha$. However, $|L_{\omega+\alpha}|=\aleph_0 + |\alpha|$ whilst $|V_{\omega+\alpha}|=\beth_\alpha$. Unless $\alpha$ is a [[Beth|$\beth$-fixed point]], $|L_{\omega+\alpha}|&lt;|V_{\omega+\alpha}|$. Although $L_\alpha$ is quite small compared to $V_\alpha$, $L$ is a tall model, meaning $L$ contains every ordinal. In fact, $V_\alpha\cap\mathrm{Ord}=L_\alpha\cap\mathrm{Ord}=\alpha$, so the ordinals in $V_\alpha$ are precisely those in $L_\alpha$. <br /> <br /> If $0^{\#}$ exists (see below), then every uncountable cardinal $\kappa$ has $L\models$&quot;$\kappa$is [[ineffable|totally ineffable]] (and therefore the smallest actually totally ineffable cardinal $\lambda$ has many more large cardinal properties in $L$). <br /> <br /> However, if $\kappa$ is [[inaccessible]] and $V=L$, then $V_\kappa=L_\kappa$. Furthermore, $V_\kappa\models (V=L)$. In the case where $V\neq L$, it is still true that $V_\kappa^L=L_\kappa$, although $V_\kappa^L$ will not be $V_\kappa$. In fact, $\mathcal{P}(\omega)\not\in V_\kappa^L$ if $0^{\#}$ exists.<br /> <br /> == Statements True in $L$ ==<br /> <br /> Here is a list of statements true in $L$:<br /> <br /> * $\text{ZFC}$ (and therefore the Axiom of Choice)<br /> * $\text{GCH}$<br /> * $V=L$ (and therefore $V$ $=$ [[HOD|$\text{HOD}$]])<br /> * The Diamond Principle <br /> * The Clubsuit Principle<br /> * The Falsity of Suslin's Hypothesis<br /> <br /> == Determinacy of $L(\R)$ ==<br /> <br /> ''Main article: [[axiom of determinacy#Determinacy of .24L.28.5Cmathbb.7BR.7D.29.24|axiom of determinacy]]''<br /> <br /> == Using other logic systems than first-order logic ==<br /> <br /> When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=\text{HOD}$. When using $\mathcal{L}_{\kappa,\kappa}$, the hierarchy is called $L_\alpha^{\mathcal{L}_{\kappa,\kappa}}$, and $L\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.<br /> <br /> Chang's Model is $L^{\mathcal{L}_{\omega_1,\omega_1}}$. Chang proved that $L^{\mathcal{L}_{\kappa,\kappa}}$ is the smallest inner model of $\text{ZFC}$ closed under sequences of length $&lt;\kappa$.<br /> <br /> == Silver indiscernibles ==<br /> <br /> ''To be expanded.''<br /> <br /> == Sharps ==<br /> <br /> $0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which, under the existence of many Silver indiscernibles (a statement independent of $\text{ZFC}$), has a certain number of properties that contredicts the [[L|axiom of constructibility]] and implies that, in short, $L$ and $V$ are &quot;''very different''&quot;. Technically, under the standard definition of $0^\#$ as a (real number encoding a) set of formulas, $0^\#$ provably exists in $\text{ZFC}$, but lacks all its important properties. Thus the expression &quot;$0^\#$ exists&quot; is to be understood as &quot;$0^\#$ exists ''and'' there are uncountably many Silver indiscernibles&quot;.<br /> <br /> === Definition of $0^{\#}$ ===<br /> <br /> Assume there is an uncountable set of Silver indiscernibles. Then $0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$.<br /> <br /> &quot;$0^{\#}$ exists&quot; is used as a shorthand for &quot;there is an uncountable set of Silver indiscernibles&quot;; since $L_{\aleph_\omega}$ is a set, $\text{ZFC}$ can define a truth predicate for it, and so the existence of $0^{\#}$ as a mere set of formulas would be trivial. It is interesting only when there are many (in fact proper class many) Silver indiscernibles. Similarly, we say that &quot;$0^{\#}$ does not exist&quot; if there are no Silver indiscernibles.<br /> <br /> === Implications, equivalences, and consequences of $0^\#$'s existence ===<br /> <br /> If $0^\#$ exists then:<br /> * $L_{\aleph_\omega}\prec L$ and so $0^\#$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$<br /> * In fact, $L_\kappa\prec L$ for every Silver indiscernible, and thus for every uncountable cardinal.<br /> * Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).<br /> * For every $\alpha\in\omega_1^L$, every Silver indiscernible (and in particular every uncountable cardinal) is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.<br /> * There are only countably many reals in $L$, i.e. $|\R\cap L|=\aleph_0$ in $V$.<br /> <br /> The following statements are equivalent:<br /> * There is an uncountable set of Silver indiscernibles (i.e. &quot;$0^\#$ exists&quot;)<br /> * There is a proper class of Silver indiscernibles (unboundedly many of them).<br /> * There is a unique well-founded remarkable E.M. set (see below).<br /> * Jensen's Covering Theorem fails (see below).<br /> * $L$ is thin, i.e. $|L\cap V_\alpha|=|\alpha|$ for all $\alpha\geq\omega$.<br /> * $\Sigma^1_1$-[[axiom of projective determinacy|determinacy]] (lightface form).<br /> * $\aleph_\omega$ is regular (hence weakly inaccessible) in $L$.<br /> * There is a nontrivial [[elementary embedding]] $j:L\to L$.<br /> * There is a proper class of nontrivial elementary embeddings $j:L\to L$.<br /> * There is a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)&lt;|\alpha|$.<br /> &lt;!--* $V_{\aleph_\omega}\cap L\models\text{ZFC}+\Pi_1^1$-replacement--&gt;<br /> <br /> The existence of $0^\#$ is implied by:<br /> * [[Chang's conjecture]]<br /> * Both $\omega_1$ and $\omega_2$ being singular (requires $\neg\text{AC}$).<br /> * The negation of the singular cardinal hypothesis ($\text{SCH}$).<br /> * The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal.<br /> * The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$.<br /> * The existence of some uncountable regular cardinal $\kappa$ such that every constructible $X\subseteq\kappa$ either contains or is disjoint from a closed unbounded set.<br /> <br /> Note that if $0^{\#}$ exists then for every Silver indiscernible (in particular for every uncountable cardinal) there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ with that indiscernible as its critical point. Thus if any such embedding exists, then a proper class of those embeddings exists.<br /> <br /> === Nonexistence of $0^\#$, Jensen's Covering Theorem ===<br /> <br /> === EM blueprints and alternative characterizations of $0^\#$ === <br /> <br /> An '''EM blueprint''' (Ehrenfeucht-Mostowski blueprint) $T$ is any theory of the form $\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$ for some ordinal $\delta&gt;\omega$ and $\alpha_0&lt;\alpha_1&lt;\alpha_2...$ are indiscernible in the structure $L_\delta$. Roughly speaking, it's the set of all true statements about $\alpha_0,\alpha_1,\alpha_2...$ in $L_\delta$.<br /> <br /> For an EM blueprint $T=\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$, '''the theory $T^{-}$''' is defined as $\{\varphi:L_\delta\models\varphi\}$ (the set of truths about any definable elements of $L_\delta$). Then, '''the structure $\mathcal{M}(T,\alpha)=(M(T,\alpha);E)\models T^{-}$''' has a very technical definition, but it is indeed uniquely (up to isomorphism) the only structure which satisfies the existence of a set $X$ of $\mathcal{M}(T,\alpha)$-ordinals such that:<br /> #$X$ is a set of indiscernibles for $\mathcal{M}(T,\alpha)$ and $(X;E)\cong\alpha$ ($X$ has order-type $\alpha$ with respect to $\mathcal{M}(T,\alpha)$)<br /> #For any formula $\varphi$ and any $x&lt;y&lt;z...$ with $x,y,z...\in X$, $\mathcal{M}(T,\alpha)\models\varphi(x,y,z...)$ iff $\mathcal{M}(T,\alpha)\models\varphi(\alpha_0,\alpha_1,\alpha_2...)$ where $\alpha_0,\alpha_1...$ are the indiscernibles used in the EM blueprint.<br /> #If $&lt;$ is an $\mathcal{M}(T,\alpha)$-definable $\mathcal{M}(T,\alpha)$-well-ordering of $\mathcal{M}(T,\alpha)$, then: $$\mathcal{M}(T,\alpha)=\{\min{}_&lt;^{\mathcal{M}(T,\alpha)}\{x:\mathcal{M}(T,\alpha)\models\varphi[x,a,b,c...]\}:\varphi\in\mathcal{L}_\in\text{ and } a,b,c...\in X\}$$<br /> <br /> $0^\#$ is then defined as the '''unique''' EM blueprint $T$ such that:<br /> #$\mathcal{M}(T,\alpha)$ is isomorphic to a transitive model $M(T,\alpha)$ of ZFC for every $\alpha$<br /> #For any infinite $\alpha$, the set of indiscernibles $X$ associated with $M(T,\alpha)$ can be made cofinal in $\text{Ord}^{M(T,\alpha)}$.<br /> #The $L_\delta$-indiscernables $\beta_0&lt;\beta_1...$ can be made so that if $&lt;$ is an $M(T,\alpha)$-definable well-ordering of $M(T,\alpha)$, then for any $(m+n+2)$-ary formula $\varphi$ such that $\min_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}&lt;\beta_m$, then: $$\min{}_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}=\min{}_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m-1},\beta_{m+n+1}...\beta_{m+2n+1}]\}$$<br /> <br /> If the EM blueprint meets 1. then it is called ''well-founded.'' If it meets 2. and 3. then it is called ''remarkable.''<br /> <br /> If $0^\#$ exists (i.e. there is a well-founded remarkable EM blueprint) then it happens to be equivalent to the set of all $\varphi$ such that $L\models\varphi[\kappa_0,\kappa_1...]$ for some uncountable cardinals $\kappa_0,\kappa_1...&lt;\aleph_\omega$. This is because the associated $M(T,\alpha)$ will always have $M(T,\alpha)\prec L$ and furthermore $\kappa_0,\kappa_1...$ would be indiscernibles for $L$.<br /> <br /> $0^\#$ exists interestingly iff some $L_\delta$ has an uncountable set of indiscernables. If $0^\#$ exists, then there is some uncountable $\delta$ such that $M(0^\#,\omega_1)=L_\delta$ and $L_\delta$ therefore has an uncountable set of indiscernables. On the other hand, if some $L_\delta$ has an uncountable set of indiscernables, then the EM blueprint of $L_\delta$ is $0^\#$.<br /> <br /> === Sharps of arbitrary sets ===<br /> <br /> == References ==<br /> *Jech, ''Thomas J. Set Theory'' (The 3rd Millennium Ed.). Springer, 2003.<br /> *user46667, ''Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)'', URL (version: 2014-03-17): https://mathoverflow.net/q/156940<br /> *Chang, C. C. (1971), &quot;Sets Constructible Using $\mathcal{L}_{\kappa,\kappa}$&quot;, ''Axiomatic Set Theory'', Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8</div> Zetapology http://cantorsattic.info/index.php?title=Constructible_universe&diff=2669 Constructible universe 2018-10-19T17:22:38Z <p>Zetapology: /* E.M. sets and alternative characterizations of $0^\#$ */</p> <hr /> <div>[[Category:Constructibility]]<br /> The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of [[ZFC|$\text{ZFC+}$]][[GCH|$\text{GCH}$]] (assuming the consistency of $\text{ZFC}$) showing that $\text{ZFC}$ cannot disprove $\text{GCH}$. It was then shown to be an important model of $\text{ZFC}$ for its satisfying of other axioms, thus making them consistent with $\text{ZFC}$. The idea is that $L$ is built up by ranks like $V$. $L_0$ is the empty set, and $L_{\alpha+1}$ is the set of all easily definable subsets of $L_\alpha$. The assumption that $V=L$ (also known as the '''Axiom of constructibility''') is undecidable from $\text{ZFC}$, and implies many axioms which are consistent with $\text{ZFC}$. A set $X$ is '''constructible''' iff $X\in L$. $V=L$ iff every set is constructible.<br /> <br /> == Definition ==<br /> <br /> $\mathrm{def}(X)$ is the set of all &quot;easily definable&quot; subsets of $X$ (specifically the $\Delta_0$ definable subsets). More specifically, a subset $x$ of $X$ is in $\mathrm{def}(X)$ iff there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,v_0,v_1...v_n]\}$. Then, $L_\alpha$ and $L$ are defined as follows:<br /> <br /> *$L_0=\emptyset$<br /> *$L_{\alpha+1}=\mathrm{def}(L_\alpha)$<br /> *$L_\beta=\bigcup_{\alpha&lt;\beta} L_\alpha$ if $\beta$ is a limit ordinal<br /> *$L=\bigcup_{\alpha\in\mathrm{Ord}} L_\alpha$<br /> <br /> === The Relativized constructible universes $L_\alpha(W)$ and $L_\alpha[W]$ ===<br /> <br /> $L_\alpha(W)$ for a class $W$ is defined the same way except $L_0(W)=\text{TC}(\{W\})$ (the transitive closure of $\{W\}$). $L_\alpha[W]$ for a class $W$ is defined in the same way as $L$ except using $\mathrm{def}_W(X)$, where $\mathrm{def}_W(X)$ is the set of all $x\subseteq X$ such that there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,W,v_0,v_1...v_n]\}$ (because the relativization of $\varphi$ to $X$ is used and $\langle X,\in\rangle$ is not used, this definition makes sense even when $W$ is not in $X$).<br /> <br /> $L[W]=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha[W]$ is always a model of $\text{ZFC}$, and always satisfies $\text{GCH}$ past a certain cardinality. $L(W)=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha(W)$ is always a model of $\text{ZF}$ but need not satisfy $\text{AC}$ (the axiom of choice). In particular, $L(\mathbb{R})$ is, under large cardinal assumptions, a model of the [[axiom of determinacy]]. However, Shelah proved that if $\lambda$ is a strong limit cardinal of uncountable cofinality then $L(\mathcal{P}(\lambda))$ is a model of $\text{AC}$.<br /> <br /> == The difference between $L_\alpha$ and $V_\alpha$ ==<br /> <br /> For $\alpha\leq\omega$, $L_\alpha=V_\alpha$. However, $|L_{\omega+\alpha}|=\aleph_0 + |\alpha|$ whilst $|V_{\omega+\alpha}|=\beth_\alpha$. Unless $\alpha$ is a [[Beth|$\beth$-fixed point]], $|L_{\omega+\alpha}|&lt;|V_{\omega+\alpha}|$. Although $L_\alpha$ is quite small compared to $V_\alpha$, $L$ is a tall model, meaning $L$ contains every ordinal. In fact, $V_\alpha\cap\mathrm{Ord}=L_\alpha\cap\mathrm{Ord}=\alpha$, so the ordinals in $V_\alpha$ are precisely those in $L_\alpha$. <br /> <br /> If $0^{\#}$ exists (see below), then every uncountable cardinal $\kappa$ has $L\models$&quot;$\kappa$is [[ineffable|totally ineffable]] (and therefore the smallest actually totally ineffable cardinal $\lambda$ has many more large cardinal properties in $L$). <br /> <br /> However, if $\kappa$ is [[inaccessible]] and $V=L$, then $V_\kappa=L_\kappa$. Furthermore, $V_\kappa\models (V=L)$. In the case where $V\neq L$, it is still true that $V_\kappa^L=L_\kappa$, although $V_\kappa^L$ will not be $V_\kappa$. In fact, $\mathcal{P}(\omega)\not\in V_\kappa^L$ if $0^{\#}$ exists.<br /> <br /> == Statements True in $L$ ==<br /> <br /> Here is a list of statements true in $L$:<br /> <br /> * $\text{ZFC}$ (and therefore the Axiom of Choice)<br /> * $\text{GCH}$<br /> * $V=L$ (and therefore $V$ $=$ [[HOD|$\text{HOD}$]])<br /> * The Diamond Principle <br /> * The Clubsuit Principle<br /> * The Falsity of Suslin's Hypothesis<br /> <br /> == Determinacy of $L(\R)$ ==<br /> <br /> ''Main article: [[axiom of determinacy#Determinacy of .24L.28.5Cmathbb.7BR.7D.29.24|axiom of determinacy]]''<br /> <br /> == Using other logic systems than first-order logic ==<br /> <br /> When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=\text{HOD}$. When using $\mathcal{L}_{\kappa,\kappa}$, the hierarchy is called $L_\alpha^{\mathcal{L}_{\kappa,\kappa}}$, and $L\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.<br /> <br /> Chang's Model is $L^{\mathcal{L}_{\omega_1,\omega_1}}$. Chang proved that $L^{\mathcal{L}_{\kappa,\kappa}}$ is the smallest inner model of $\text{ZFC}$ closed under sequences of length $&lt;\kappa$.<br /> <br /> == Silver indiscernibles ==<br /> <br /> ''To be expanded.''<br /> <br /> == Sharps ==<br /> <br /> $0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which, under the existence of many Silver indiscernibles (a statement independent of $\text{ZFC}$), has a certain number of properties that contredicts the [[L|axiom of constructibility]] and implies that, in short, $L$ and $V$ are &quot;''very different''&quot;. Technically, under the standard definition of $0^\#$ as a (real number encoding a) set of formulas, $0^\#$ provably exists in $\text{ZFC}$, but lacks all its important properties. Thus the expression &quot;$0^\#$ exists&quot; is to be understood as &quot;$0^\#$ exists ''and'' there are uncountably many Silver indiscernibles&quot;.<br /> <br /> === Definition of $0^{\#}$ ===<br /> <br /> Assume there is an uncountable set of Silver indiscernibles. Then $0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$.<br /> <br /> &quot;$0^{\#}$ exists&quot; is used as a shorthand for &quot;there is an uncountable set of Silver indiscernibles&quot;; since $L_{\aleph_\omega}$ is a set, $\text{ZFC}$ can define a truth predicate for it, and so the existence of $0^{\#}$ as a mere set of formulas would be trivial. It is interesting only when there are many (in fact proper class many) Silver indiscernibles. Similarly, we say that &quot;$0^{\#}$ does not exist&quot; if there are no Silver indiscernibles.<br /> <br /> === Implications, equivalences, and consequences of $0^\#$'s existence ===<br /> <br /> If $0^\#$ exists then:<br /> * $L_{\aleph_\omega}\prec L$ and so $0^\#$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$<br /> * In fact, $L_\kappa\prec L$ for every Silver indiscernible, and thus for every uncountable cardinal.<br /> * Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).<br /> * For every $\alpha\in\omega_1^L$, every Silver indiscernible (and in particular every uncountable cardinal) is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.<br /> * There are only countably many reals in $L$, i.e. $|\R\cap L|=\aleph_0$ in $V$.<br /> <br /> The following statements are equivalent:<br /> * There is an uncountable set of Silver indiscernibles (i.e. &quot;$0^\#$ exists&quot;)<br /> * There is a proper class of Silver indiscernibles (unboundedly many of them).<br /> * There is a unique well-founded remarkable E.M. set (see below).<br /> * Jensen's Covering Theorem fails (see below).<br /> * $L$ is thin, i.e. $|L\cap V_\alpha|=|\alpha|$ for all $\alpha\geq\omega$.<br /> * $\Sigma^1_1$-[[axiom of projective determinacy|determinacy]] (lightface form).<br /> * $\aleph_\omega$ is regular (hence weakly inaccessible) in $L$.<br /> * There is a nontrivial [[elementary embedding]] $j:L\to L$.<br /> * There is a proper class of nontrivial elementary embeddings $j:L\to L$.<br /> * There is a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)&lt;|\alpha|$.<br /> &lt;!--* $V_{\aleph_\omega}\cap L\models\text{ZFC}+\Pi_1^1$-replacement--&gt;<br /> <br /> The existence of $0^\#$ is implied by:<br /> * [[Chang's conjecture]]<br /> * Both $\omega_1$ and $\omega_2$ being singular (requires $\neg\text{AC}$).<br /> * The negation of the singular cardinal hypothesis ($\text{SCH}$).<br /> * The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal.<br /> * The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$.<br /> * The existence of some uncountable regular cardinal $\kappa$ such that every constructible $X\subseteq\kappa$ either contains or is disjoint from a closed unbounded set.<br /> <br /> Note that if $0^{\#}$ exists then for every Silver indiscernible (in particular for every uncountable cardinal) there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ with that indiscernible as its critical point. Thus if any such embedding exists, then a proper class of those embeddings exists.<br /> <br /> === Nonexistence of $0^\#$, Jensen's Covering Theorem ===<br /> <br /> === EM blueprints and alternative characterizations of $0^\#$ === <br /> <br /> An '''EM blueprint''' (Ehrenfeucht-Mostowski blueprint) $T$ is any theory of the form $\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$ for some ordinal $\delta&gt;\omega$ and $\alpha_0&lt;\alpha_1&lt;\alpha_2...$ are indiscernible in the structure $L_\delta$. Roughly speaking, it's the set of all true statements about $\alpha_0,\alpha_1,\alpha_2...$ in $L_\delta$.<br /> <br /> For an EM blueprint $T=\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$, '''the theory $T^{-}$''' is defined as $\{\varphi:L_\delta\models\varphi\}$ (the set of truths about any definable elements of $L_\delta$). Then, '''the structure $\mathcal{M}(T,\alpha)=(M(T,\alpha);E)\models T^{-}$''' has a very technical definition, but it is indeed uniquely (up to isomorphism) the only structure which satisfies the existence of a set $X$ of $\mathcal{M}(T,\alpha)$-ordinals such that:<br /> #$X$ is a set of indiscernibles for $\mathcal{M}(T,\alpha)$ and $(X;E)\cong\alpha$ ($X$ has order-type $\alpha$ with respect to $\mathcal{M}(T,\alpha)$)<br /> #For any formula $\varphi$ and any $x&lt;y&lt;z...$ with $x,y,z...\in X$, $\mathcal{M}(T,\alpha)\models\varphi(x,y,z...)$ iff $\mathcal{M}(T,\alpha)\models\varphi(\alpha_0,\alpha_1,\alpha_2...)$ where $\alpha_0,\alpha_1...$ are the indiscernibles used in the EM blueprint.<br /> #If $&lt;$ is an $\mathcal{M}(T,\alpha)$-definable $\mathcal{M}(T,\alpha)$-well-ordering of $\mathcal{M}(T,\alpha)$, then: $$\mathcal{M}(T,\alpha)=\{\min{}_&lt;^{\mathcal{M}(T,\alpha)}\{x:\mathcal{M}(T,\alpha)\models\varphi[x,a,b,c...]\}:\varphi\in\mathcal{L}_\in\text{ and } a,b,c...\in X\}$$<br /> <br /> $0^\#$ is then defined as the '''unique''' EM blueprint $T$ such that:<br /> #$\mathcal{M}(T,\alpha)$ is isomorphic to a transitive model $M(T,\alpha)$ of ZFC for every $\alpha$<br /> #For any infinite $\alpha$, the set of indiscernibles $X$ associated with $M(T,\alpha)$ can be made cofinal in $\text{Ord}^{M(T,\alpha)}$.<br /> #The $L_\delta$-indiscernables $\beta_0&lt;\beta_1...$ can be made so that if $&lt;$ is an $M(T,\alpha)$-definable well-ordering of $M(T,\alpha)$, then for any $(m+n+2)$-ary formula $\varphi$ such that $\min_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}&lt;\beta_m$, then: $$\min{}_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}=\min{}_&lt;^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m-1},\beta_{m+n+1}...\beta_{m+2n+1}]\}$$<br /> <br /> If $0^\#$ exists (i.e. there is such an EM blueprint) then it happens to be equivalent to the set of all $\varphi$ such that $L\models\varphi[\kappa_0,\kappa_1...]$ for some uncountable cardinals $\kappa_0,\kappa_1...&lt;\aleph_\omega$.<br /> <br /> === Sharps of arbitrary sets ===<br /> <br /> == References ==<br /> *Jech, ''Thomas J. Set Theory'' (The 3rd Millennium Ed.). Springer, 2003.<br /> *user46667, ''Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)'', URL (version: 2014-03-17): https://mathoverflow.net/q/156940<br /> *Chang, C. C. (1971), &quot;Sets Constructible Using $\mathcal{L}_{\kappa,\kappa}$&quot;, ''Axiomatic Set Theory'', Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2665 Tall 2018-10-11T02:40:13Z <p>Zetapology: /* Embedding Characterization */ tightened characterization</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ 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 $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$. 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 $\beta&lt;\alpha$ 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 $\beta&lt;\alpha$ 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> Zetapology http://cantorsattic.info/index.php?title=Upper_attic&diff=2664 Upper attic 2018-10-10T17:55:50Z <p>Zetapology: </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 /> * The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal<br /> * '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings<br /> * The [[wholeness axioms]]<br /> * [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...<br /> * '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[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 cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal<br /> * [[Vopenka | Vopěnka's principle]]<br /> * [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal<br /> &lt;!--* [[grand reflection]] cardinal--&gt;<br /> * [[hypercompact]] cardinal<br /> * '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal<br /> * '''[[strongly compact]]''' cardinal [[strongly compact | $\lambda$-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<br /> * [[Woodin#Shelah|Shelah]] cardinal<br /> * The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br /> * '''[[Woodin]]''' cardinal<br /> * [[strongly tall]] cardinal<br /> * [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy<br /> * Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br /> *[[zero dagger| $0^\dagger$]] (''zero-dagger'')<br /> * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal<br /> * [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal<br /> * [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal<br /> * '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] 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<br /> * '''[[zero sharp | $0^\sharp$]]''' (''zero-sharp''), existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]<br /> * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br /> * the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq\alpha&lt;\omega_1$<br /> * [[remarkable]] cardinal<br /> * [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal<br /> * [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal<br /> * [[subtle]] cardinal<br /> * [[ineffable#Ethereal cardinal|ethereal]] cardinal<br /> * [[unfoldable#Superstrongly Unfoldable | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br /> * [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br /> * [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable | strongly unfoldable]] cardinal<br /> * [[indescribable]] hierarchy, [[totally indescribable]] cardinal<br /> * '''[[weakly compact]]''' cardinal<br /> * The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$ <br /> * '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals<br /> * [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal<br /> * [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<br /> * [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] 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 and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals<br /> * [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals<br /> * '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly 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 | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]<br /> * '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br /> <br /> * down to [[the middle attic]]</div> Zetapology http://cantorsattic.info/index.php?title=Strongly_tall&diff=2663 Strongly tall 2018-10-10T17:55:15Z <p>Zetapology: added redirect page</p> <hr /> <div>#REDIRECT [[tall#Strongly Tall Cardinals]]</div> Zetapology http://cantorsattic.info/index.php?title=Strongly_compact&diff=2662 Strongly compact 2018-10-10T17:53:12Z <p>Zetapology: /* Relation to other large cardinal notions */ relation to strongly tall cardinals</p> <hr /> <div>{{DISPLAYTITLE: Strongly compact cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an<br /> uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$. <br /> <br /> The strongly compact embedding characterizations are closely related to that of [[supercompact]] cardinals, which are characterized by [[elementary embedding|elementary embeddings]] with a high degree of closure: $\kappa$ is $\theta$-[[supercompact]] if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta&lt;j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-[[supercompact]] cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\subset M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.<br /> <br /> == Diverse characterizations ==<br /> <br /> There are diverse equivalent characterizations of the strongly compact cardinals. <br /> <br /> === Strong compactness characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha&lt;\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha&lt;\delta}\varphi_\alpha$ of any size $\delta&lt;\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\langle x_\alpha\mid\alpha&lt;\delta\rangle$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.<br /> <br /> === Strong compactness embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Cover property characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\subset M$ of size $\theta$ there is $s\in M$ with $t\subset s$ and $|s|^M&lt;j(\kappa)$.<br /> <br /> === Fine measure characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a [[filter|fine measure]] on $\mathcal{P}_\kappa(\theta)$. The notation $\mathcal{P}_\kappa(\theta)$ means $\{\sigma\subset\theta\mid |\sigma|&lt;\kappa\}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Filter extension characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete [[filter]] of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Discontinuous ultrapower characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda&lt;j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$. <br /> <br /> === Discontinuous embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda&lt;j(\lambda)$.<br /> <br /> === Ketonen characterization ===<br /> <br /> An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha&lt;\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality. <br /> <br /> === Regular ultrafilter characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\{X_\alpha\mid\alpha&lt;\theta\}\subset \mu$ such that $\bigcap_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.<br /> <br /> == Strongly compact cardinals and forcing ==<br /> <br /> If there is proper class-many strongly compact cardinals, then there is a [[forcing|generic model]] of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot;. If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot; + &quot;every uncountable cardinal is both almost [[Ramsey]] and a [[Rowbottom]] cardinal carrying a Rowbottom filter&quot;. <br /> This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals.<br /> <br /> == Relation to other large cardinal notions ==<br /> <br /> Strongly compact cardinals are [[measurable]]. The least strongly compact cardinal can be equal to the least measurable cardinal, or to the least [[supercompact]] cardinal, by results of Magidor. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)<br /> <br /> Even though strongly compact cardinals imply the consistency of the negation of the singular cardinal hypothesis, SCH, for any singular strong limit cardinal $\kappa$ above the least strongly compact cardinal, $2^\kappa=\kappa^+$ (also known as &quot;SCH holds above strong compactness&quot;). &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> If there is a strongly compact cardinal $\kappa$ then for all $\lambda\geq\kappa$ and $A\subseteq\lambda$, $\lambda^+$ is [[ineffable]] in $L[A]$.<br /> <br /> 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.<br /> <br /> Every strongly compact cardinal is [[strongly tall]], although the existence of a strongly compact cardinal is equiconsistent with &quot;the least measurable cardinal is the least strongly compact cardinal, and therefore the least strongly tall cardinal&quot;, so it could be the case that the least of the measurable, tall, strongly tall, and strongly compact cardinals all line up.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2661 Tall 2018-10-10T17:49:12Z <p>Zetapology: /* Strongly Tall Cardinals */ added embedding characterization</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ 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 $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$. 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 there is some elementary embedding $j:V\prec M$, a set $S$ and an $A\in j(S)$ such that:<br /> #For any $\alpha\leq\theta$, there is a function $f$ with domain $S$ such that $j(f)(A)=\alpha$.<br /> #$j(\kappa)&gt;\theta$.<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 $\beta&lt;\alpha$ 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 $\beta&lt;\alpha$ 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> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2654 Tall 2018-10-08T07:26:56Z <p>Zetapology: /* Ultrafilter Characterization */</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ 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 $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<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 $H_\alpha:S\rightarrow\text{Ord}$ 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 $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=H_\beta(x)\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=H_\beta(x)$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2653 Tall 2018-10-08T03:23:39Z <p>Zetapology: /* Strongly Tall Cardinals */ completeness $\kappa^+$ -&gt; $\kappa$-complete</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ 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 $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<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 $H_\alpha:S\rightarrow\text{Ord}$ 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 $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=\beta$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Huge&diff=2652 Huge 2018-10-08T02:42:23Z <p>Zetapology: /* Consistency strength and size */ added relation to strong cardinals</p> <hr /> <div>{{DISPLAYTITLE: Huge cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> '''Huge''' cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+&quot;there is a $\omega_2$-saturated $\sigma$-[[filter|ideal]] on $\omega_1$&quot;. It is now known that only a [[Woodin]] cardinal is needed for this result. However, the consistency of the existence of an $\omega_2$-complete $\omega_3$-saturated $\sigma$-ideal on $\omega_2$, as far as the set theory world is concerned, still requires an almost huge cardinal. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> == Definitions ==<br /> <br /> Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the &quot;double helix&quot;, in which for some large cardinal properties n-$P_0$ and n-$P_1$, n-$P_0$ has less consistency strength than n-$P_1$, which has less consistency strength than (n+1)-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|n-fold variants]] as of modern set theoretic concerns. &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt;<br /> <br /> Although they are very large, there is a first-order definition which is equivalent to n-hugeness, so the $\theta$-th n-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br /> <br /> === Elementary embedding definitions ===<br /> <br /> The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be a nontrivial [[elementary embedding]] of $V$ into a [[transitive]] class $M$ with critical point $\kappa$. Then:<br /> <br /> *$\kappa$ is '''almost n-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{&lt;\lambda}\subseteq M$).<br /> *$\kappa$ is '''n-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subseteq M$).<br /> *$\kappa$ is '''almost n-huge''' iff it is almost n-huge with target $\lambda$ for some $\lambda$.<br /> *$\kappa$ is '''n-huge''' iff it is n-huge with target $\lambda$ for some $\lambda$.<br /> *$\kappa$ is '''super almost n-huge''' iff for every $\gamma$, there is some $\lambda&gt;\gamma$ for which $\kappa$ is almost n-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br /> *$\kappa$ is '''super n-huge''' iff for every $\gamma$, there is some $\lambda&gt;\gamma$ for which $\kappa$ is n-huge with target $\lambda$.<br /> *$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost 1-huge''', '''1-huge''', etc. respectively.<br /> <br /> === Ultrahuge cardinals ===<br /> <br /> A cardinal $\kappa$ is '''$\lambda$-ultrahuge''' for $\lambda&gt;\kappa$ if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{j}(\kappa)&gt;\lambda$, $M^{j(\kappa)}\subseteq M$ and $V_{j(\lambda)}\subseteq M$. A cardinal is '''ultrahuge''' if it is $\lambda$-ultrahuge for all $\lambda\geq\kappa$. [http://logicatorino.altervista.org/slides/150619tsaprounis.pdf] Notice how similar this definition is to the alternative characterization of [[extendible]] cardinals. Furthermore, this definition can be extended in the obvious way to define $\lambda$-ultra n-hugeness and ultra n-hugeness, as well as the &quot;''almost''&quot; variants.<br /> <br /> === Ultrafilter definition ===<br /> <br /> The first-order definition of n-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is n-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0&lt;\lambda_1&lt;\lambda_2...&lt;\lambda_{n-1}&lt;\lambda_n=\lambda$ such that:<br /> <br /> $$\forall i&lt;n(\{x\subseteq\lambda:\text{order-type}(x\cap\lambda_{i+1})=\lambda_i\}\in U)$$<br /> <br /> Where $\text{order-type}(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; $\kappa$ is then super n-huge if for all ordinals $\theta$ there is a $\lambda&gt;\theta$ such that $\kappa$ is n-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses n-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.<br /> <br /> As an example, $\kappa$ is 1-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$ such that $\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}\in U$. The reason why this would be so surprising is that every set $x\subseteq\lambda$ with every set of order-type $\kappa$ would be in the ultrafilter; that is, every set containing $\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}$ as a subset is considered a &quot;large set.&quot;<br /> <br /> == Consistency strength and size ==<br /> <br /> Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|n-fold variants]]) known as the ''double helix''. This phenomenon is when for one n-fold variant, letting a cardinal be called n-$P_0$ iff it has the property, and another variant, n-$P_1$, n-$P_0$ is weaker than n-$P_1$, which is weaker than (n+1)-$P_0$. &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt; In the consistency strength hierarchy, here is where these lay (top being weakest):<br /> <br /> * [[measurable]] = 0-[[superstrong]] = 0-huge<br /> * n-superstrong<br /> * n-fold supercompact<br /> * (n+1)-fold strong, n-fold extendible<br /> * (n+1)-fold Woodin, n-fold Vopěnka<br /> * (n+1)-fold Shelah<br /> * almost n-huge<br /> * super almost n-huge<br /> * n-huge<br /> * super n-huge<br /> * ultra n-huge<br /> * (n+1)-superstrong<br /> <br /> All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] n, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of n-huge cardinals, for all n. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda&lt;\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda&lt;\kappa$. Every (n+1)-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$&quot;$\lambda$ is super n-huge&quot; &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;, in fact it contains every cardinal $\lambda$ such that $V_\kappa\models$&quot;$\lambda$ is ultra n-huge&quot;.<br /> <br /> Every n-huge cardinal is m-huge for every m&lt;n. Similarly with almost n-hugeness, super n-hugeness, and super almost n-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; Every ultra n-huge is super n-huge and a stationary limit of super n-huge cardinals. Every super almost (n+1)-huge is ultra n-huge and a stationary limit of ultra n-huge cardinals.<br /> <br /> In terms of size, however, the least n-huge cardinal is smaller than the least [[supercompact]] cardinal (assuming both exist). &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; This is because n-huge cardinals have upward reflection properties, while supercompacts have downward reflection properties. Thus for any $\kappa$ which is supercompact and has an n-huge cardinal above it, $\kappa$ &quot;reflects downward&quot; that n-huge cardinal: there are $\kappa$-many n-huge cardinals below $\kappa$. On the other hand, the least super n-huge cardinals have ''both'' upward and downward reflection properties, and are all ''much'' larger than the least supercompact cardinal. It is notable that, while almost 2-huge cardinals have higher consistency strength than superhuge cardinals, the least almost 2-huge is much smaller than the least super almost huge.<br /> <br /> While not every $n$-huge cardinal is [[strong]], if $\kappa$ is almost $n$-huge with targets $\lambda_1,\lambda_2...\lambda_n$, then $\kappa$ is $\lambda_n$-strong as witnessed by the generated $j:V\prec M$. This is because $j^n(\kappa)=\lambda_n$ is [[measurable]] and therefore $\beth_{\lambda_n}=\lambda_n$ and so $V_{\lambda_n}=H_{\lambda_n}$ and because $M^{&lt;\lambda_n}\subset M$, $H_\theta\subset M$ for each $\theta&lt;\lambda_n$ and so $\cup\{H_\theta:\theta&lt;\lambda_n\} = \cup\{V_\theta:\theta&lt;\lambda_n\} = V_{\lambda_n}\subset M$.<br /> <br /> Every almost $n$-huge cardinal with targets $\lambda_1,\lambda_2...\lambda_n$ is also [[supercompact|$\theta$-supercompact]] for each $\theta&lt;\lambda_n$, and every $n$-huge cardinal with targets $\lambda_1,\lambda_2...\lambda_n$ is also $\lambda_n$-supercompact.<br /> <br /> === $\omega$-Huge Cardinals ===<br /> <br /> A cardinal $\kappa$ is '''almost $\omega$-huge''' iff there is some transitive model $M$ and an elementary embedding $j:V\prec M$ with critical point $\kappa$ such that $M^{&lt;\lambda}\subset M$ where $\lambda$ is the smallest cardinal above $\kappa$ such that $j(\lambda)=\lambda$. Similarly, $\kappa$ is '''$\omega$-huge''' iff the model $M$ can be required to have $M^\lambda\subset M$. <br /> <br /> Sadly, $\omega$-huge cardinals are inconsistent with ZFC by a version of Kunen's inconsistency theorem. Now, $\omega$-hugeness is used to describe critical points of [[rank-into-rank|I1 embeddings]].<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Supercompact&diff=2651 Supercompact 2018-10-08T02:36:32Z <p>Zetapology: /* Relation to other large cardinals */ more general result</p> <hr /> <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 /> <br /> ==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 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 $\eta&gt;\kappa$ there is $\alpha&lt;\kappa$ such that there exists a nontrivial elementary embedding $j:V_\alpha\to V_\eta$ such that $j(\mathrm{crit}(j))=\kappa$.<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 $2^{2^{\lambda^{&lt;\kappa}}}$ normal fine measures on $\mathcal{P}_\kappa(\lambda)$.<br /> <br /> Every supercompact has [[Mitchell order]] $(2^\kappa)^+\geq\kappa^{++}$.<br /> <br /> If $\kappa$ is $\lambda$-supercompact then it is also $\mu$-supercompact for every $\mu&lt;\lambda$. If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha&lt;\kappa$ that is $\gamma$-supercompact for all $\gamma&lt;\kappa$ (if any exists) is also $\lambda$-supercompact. In the same vein, for every cardinals $\kappa&lt;\lambda$, if $\lambda$ is supercompact and $\kappa$ is $\gamma$-supercompact for all $\gamma&lt;\lambda$, 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 $|tc(x)|\leq\lambda$ there exists a normal fine measure $U$ on $\mathcal{P}_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the ''transitive closure'' of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.<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 $\alpha&lt;\kappa$, 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 $2^{\aleph_\alpha}&lt;\aleph_{\omega_1}$ for all $\alpha&lt;\omega_1$, but also $2^{\aleph_{\omega_1}}&gt;\aleph_{\omega_1+\alpha+1}$ for any prescribed $\alpha&lt;\omega_2$. Similarly, one can have a generic extension in which the $\text{GCH}$ holds below $\aleph_\omega$ but $2^{\aleph_\omega}&gt;\aleph_{\omega+\alpha+1}$ for any prescribed $\alpha&lt;\omega_1$.<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.''<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 &lt;$\Theta$-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 $\eta&lt;\kappa$ 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 /> {{stub}}</div> Zetapology http://cantorsattic.info/index.php?title=Supercompact&diff=2650 Supercompact 2018-10-08T02:26:24Z <p>Zetapology: /* Relation to other large cardinals */ relation to strong cardinals</p> <hr /> <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 /> <br /> ==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 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 $\eta&gt;\kappa$ there is $\alpha&lt;\kappa$ such that there exists a nontrivial elementary embedding $j:V_\alpha\to V_\eta$ such that $j(\mathrm{crit}(j))=\kappa$.<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 $2^{2^{\lambda^{&lt;\kappa}}}$ normal fine measures on $\mathcal{P}_\kappa(\lambda)$.<br /> <br /> Every supercompact has [[Mitchell order]] $(2^\kappa)^+\geq\kappa^{++}$.<br /> <br /> If $\kappa$ is $\lambda$-supercompact then it is also $\mu$-supercompact for every $\mu&lt;\lambda$. If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha&lt;\kappa$ that is $\gamma$-supercompact for all $\gamma&lt;\kappa$ (if any exists) is also $\lambda$-supercompact. In the same vein, for every cardinals $\kappa&lt;\lambda$, if $\lambda$ is supercompact and $\kappa$ is $\gamma$-supercompact for all $\gamma&lt;\lambda$, 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 $|tc(x)|\leq\lambda$ there exists a normal fine measure $U$ on $\mathcal{P}_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the ''transitive closure'' of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.<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 $\alpha&lt;\kappa$, 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 $2^{\aleph_\alpha}&lt;\aleph_{\omega_1}$ for all $\alpha&lt;\omega_1$, but also $2^{\aleph_{\omega_1}}&gt;\aleph_{\omega_1+\alpha+1}$ for any prescribed $\alpha&lt;\omega_2$. Similarly, one can have a generic extension in which the $\text{GCH}$ holds below $\aleph_\omega$ but $2^{\aleph_\omega}&gt;\aleph_{\omega+\alpha+1}$ for any prescribed $\alpha&lt;\omega_1$.<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.''<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 /> Let $B_\theta$ be the $\theta$-th cardinal $\lambda$ such that $\lambda=\beth_\lambda$. Then, every $B_\theta$-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. As a result, if $\theta=B_\theta$ (the least such cardinal has [[cofinality]] $\omega$), then every $\theta$-supercompact cardinal is $\theta$-strong.<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 &lt;$\Theta$-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 $\eta&lt;\kappa$ 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 /> {{stub}}</div> Zetapology http://cantorsattic.info/index.php?title=Huge&diff=2649 Huge 2018-10-07T20:04:50Z <p>Zetapology: /* $\omega$-Huge Cardinals */ almost w-huge cardinals not known to be inconsistent (?)</p> <hr /> <div>{{DISPLAYTITLE: Huge cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> '''Huge''' cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+&quot;there is a $\omega_2$-saturated $\sigma$-[[filter|ideal]] on $\omega_1$&quot;. It is now known that only a [[Woodin]] cardinal is needed for this result. However, the consistency of the existence of an $\omega_2$-complete $\omega_3$-saturated $\sigma$-ideal on $\omega_2$, as far as the set theory world is concerned, still requires an almost huge cardinal. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> == Definitions ==<br /> <br /> Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the &quot;double helix&quot;, in which for some large cardinal properties n-$P_0$ and n-$P_1$, n-$P_0$ has less consistency strength than n-$P_1$, which has less consistency strength than (n+1)-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|n-fold variants]] as of modern set theoretic concerns. &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt;<br /> <br /> Although they are very large, there is a first-order definition which is equivalent to n-hugeness, so the $\theta$-th n-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br /> <br /> === Elementary embedding definitions ===<br /> <br /> The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be a nontrivial [[elementary embedding]] of $V$ into a [[transitive]] class $M$ with critical point $\kappa$. Then:<br /> <br /> *$\kappa$ is '''almost n-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{&lt;\lambda}\subseteq M$).<br /> *$\kappa$ is '''n-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subseteq M$).<br /> *$\kappa$ is '''almost n-huge''' iff it is almost n-huge with target $\lambda$ for some $\lambda$.<br /> *$\kappa$ is '''n-huge''' iff it is n-huge with target $\lambda$ for some $\lambda$.<br /> *$\kappa$ is '''super almost n-huge''' iff for every $\gamma$, there is some $\lambda&gt;\gamma$ for which $\kappa$ is almost n-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br /> *$\kappa$ is '''super n-huge''' iff for every $\gamma$, there is some $\lambda&gt;\gamma$ for which $\kappa$ is n-huge with target $\lambda$.<br /> *$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost 1-huge''', '''1-huge''', etc. respectively.<br /> <br /> === Ultrahuge cardinals ===<br /> <br /> A cardinal $\kappa$ is '''$\lambda$-ultrahuge''' for $\lambda&gt;\kappa$ if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{j}(\kappa)&gt;\lambda$, $M^{j(\kappa)}\subseteq M$ and $V_{j(\lambda)}\subseteq M$. A cardinal is '''ultrahuge''' if it is $\lambda$-ultrahuge for all $\lambda\geq\kappa$. [http://logicatorino.altervista.org/slides/150619tsaprounis.pdf] Notice how similar this definition is to the alternative characterization of [[extendible]] cardinals. Furthermore, this definition can be extended in the obvious way to define $\lambda$-ultra n-hugeness and ultra n-hugeness, as well as the &quot;''almost''&quot; variants.<br /> <br /> === Ultrafilter definition ===<br /> <br /> The first-order definition of n-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is n-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0&lt;\lambda_1&lt;\lambda_2...&lt;\lambda_{n-1}&lt;\lambda_n=\lambda$ such that:<br /> <br /> $$\forall i&lt;n(\{x\subseteq\lambda:\text{order-type}(x\cap\lambda_{i+1})=\lambda_i\}\in U)$$<br /> <br /> Where $\text{order-type}(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; $\kappa$ is then super n-huge if for all ordinals $\theta$ there is a $\lambda&gt;\theta$ such that $\kappa$ is n-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses n-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.<br /> <br /> As an example, $\kappa$ is 1-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$ such that $\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}\in U$. The reason why this would be so surprising is that every set $x\subseteq\lambda$ with every set of order-type $\kappa$ would be in the ultrafilter; that is, every set containing $\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}$ as a subset is considered a &quot;large set.&quot;<br /> <br /> == Consistency strength and size ==<br /> <br /> Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|n-fold variants]]) known as the ''double helix''. This phenomenon is when for one n-fold variant, letting a cardinal be called n-$P_0$ iff it has the property, and another variant, n-$P_1$, n-$P_0$ is weaker than n-$P_1$, which is weaker than (n+1)-$P_0$. &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt; In the consistency strength hierarchy, here is where these lay (top being weakest):<br /> <br /> * [[measurable]] = 0-[[superstrong]] = 0-huge<br /> * n-superstrong<br /> * n-fold supercompact<br /> * (n+1)-fold strong, n-fold extendible<br /> * (n+1)-fold Woodin, n-fold Vopěnka<br /> * (n+1)-fold Shelah<br /> * almost n-huge<br /> * super almost n-huge<br /> * n-huge<br /> * super n-huge<br /> * ultra n-huge<br /> * (n+1)-superstrong<br /> <br /> All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] n, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of n-huge cardinals, for all n. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda&lt;\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda&lt;\kappa$. Every (n+1)-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$&quot;$\lambda$ is super n-huge&quot; &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;, in fact it contains every cardinal $\lambda$ such that $V_\kappa\models$&quot;$\lambda$ is ultra n-huge&quot;.<br /> <br /> Every n-huge cardinal is m-huge for every m&lt;n. Similarly with almost n-hugeness, super n-hugeness, and super almost n-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; Every ultra n-huge is super n-huge and a stationary limit of super n-huge cardinals. Every super almost (n+1)-huge is ultra n-huge and a stationary limit of ultra n-huge cardinals.<br /> <br /> In terms of size, however, the least n-huge cardinal is smaller than the least [[supercompact]] cardinal (assuming both exist). &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt; This is because n-huge cardinals have upward reflection properties, while supercompacts have downward reflection properties. Thus for any $\kappa$ which is supercompact and has an n-huge cardinal above it, $\kappa$ &quot;reflects downward&quot; that n-huge cardinal: there are $\kappa$-many n-huge cardinals below $\kappa$. On the other hand, the least super n-huge cardinals have ''both'' upward and downward reflection properties, and are all ''much'' larger than the least supercompact cardinal. It is notable that, while almost 2-huge cardinals have higher consistency strength than superhuge cardinals, the least almost 2-huge is much smaller than the least super almost huge.<br /> <br /> === $\omega$-Huge Cardinals ===<br /> <br /> A cardinal $\kappa$ is '''almost $\omega$-huge''' iff there is some transitive model $M$ and an elementary embedding $j:V\prec M$ with critical point $\kappa$ such that $M^{&lt;\lambda}\subset M$ where $\lambda$ is the smallest cardinal above $\kappa$ such that $j(\lambda)=\lambda$. Similarly, $\kappa$ is '''$\omega$-huge''' iff the model $M$ can be required to have $M^\lambda\subset M$. <br /> <br /> Sadly, $\omega$-huge cardinals are inconsistent with ZFC by a version of Kunen's inconsistency theorem. Now, $\omega$-hugeness is used to describe critical points of [[rank-into-rank|I1 embeddings]].<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2648 Tall 2018-10-07T19:29:12Z <p>Zetapology: /* Strongly Tall Cardinals */ Consistency strength is known</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ 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 an [[filter|ultrafilter]] $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<br /> <br /> === Ultrafilter Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff there is some set $S$, an [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:<br /> #$\kappa$ is uncountable.<br /> #$U$ has completeness $\kappa^+$.<br /> #$H_0(x)=0$ for each $x\in S$.<br /> #For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=\beta$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Upper_attic&diff=2647 Upper attic 2018-10-07T19:20:38Z <p>Zetapology: consistency strength of strongly tall cardinals is known</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 /> * The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal<br /> * '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings<br /> * The [[wholeness axioms]]<br /> * [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...<br /> * '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[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 cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal<br /> * [[Vopenka | Vopěnka's principle]]<br /> * [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal<br /> &lt;!--* [[grand reflection]] cardinal--&gt;<br /> * [[hypercompact]] cardinal<br /> * '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal<br /> * '''[[strongly compact]]''' cardinal [[strongly compact | $\lambda$-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<br /> * [[Woodin#Shelah|Shelah]] cardinal<br /> * The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br /> * '''[[Woodin]]''' cardinal<br /> * [[tall|strongly tall]] cardinals<br /> * [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy<br /> * Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br /> *[[zero dagger| $0^\dagger$]] (''zero-dagger'')<br /> * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal<br /> * [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal<br /> * [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal<br /> * '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] 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<br /> * '''[[zero sharp | $0^\sharp$]]''' (''zero-sharp''), existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]<br /> * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br /> * the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq\alpha&lt;\omega_1$<br /> * [[remarkable]] cardinal<br /> * [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal<br /> * [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal<br /> * [[subtle]] cardinal<br /> * [[ineffable#Ethereal cardinal|ethereal]] cardinal<br /> * [[unfoldable#Superstrongly Unfoldable | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br /> * [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br /> * [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable | strongly unfoldable]] cardinal<br /> * [[indescribable]] hierarchy, [[totally indescribable]] cardinal<br /> * '''[[weakly compact]]''' cardinal<br /> * The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$ <br /> * '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals<br /> * [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal<br /> * [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<br /> * [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] 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 and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals<br /> * [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals<br /> * '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly 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 | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]<br /> * '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br /> <br /> * down to [[the middle attic]]</div> Zetapology http://cantorsattic.info/index.php?title=Jonsson&diff=2646 Jonsson 2018-10-07T18:14:04Z <p>Zetapology: /* Embedding Characterization */</p> <hr /> <div>{{DISPLAYTITLE: Jónsson cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Partition property]]<br /> Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called '''Jónsson cardinals'''.<br /> <br /> An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of that algebra is a set $B\subseteq A$ which $B$ is closed under each $f_k$ for $k\leq n$.<br /> <br /> == Equivalent Definitions ==<br /> <br /> There are several equivalent definitions of Jónsson cardinals.<br /> <br /> === Partition Property ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff the [[partition property]] $\kappa\rightarrow [\kappa]_\kappa^{&lt;\omega}$ holds, i.e. that given any function $f:[\kappa]^{&lt;\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ of order type $\kappa$ such that $f[H]^n\neq\kappa$ for every $n&lt;\omega$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Substructure Characterization ===<br /> *A cardinal $\kappa$ is '''Jónsson''' iff given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$. <br /> <br /> *A cardinal $\kappa$ is '''Jónsson''' iff any structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Embedding Characterization ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff for every $\theta&gt;\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j&lt;\kappa$, iff for every $\theta&gt;\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to V_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit} j&lt;\kappa$.<br /> <br /> Interestingly, if one such $\theta&gt;\kappa$ has this property, then every $\theta&gt;\kappa$ has this property as well.<br /> <br /> === Abstract Algebra Characterization ===<br /> <br /> In terms of abstract algebra, $\kappa$ is '''Jónsson''' iff any algebra $A$ of size $\kappa$ has a proper subalgebra of $A$ of size $\kappa$.<br /> <br /> == Properties ==<br /> <br /> All the following facts can be found in &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;:<br /> <br /> * $\aleph_0$ is not Jónsson.<br /> * If $\kappa$ isn't Jónsson then neither is $\kappa^+$.<br /> * If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.<br /> * If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).<br /> * A singular limit of [[measurable|measurables]] is Jónsson.<br /> * The least Jónsson is either [[inaccessible|weakly inaccessible]] or has cofinality $\omega$.<br /> * $\aleph_{\omega+1}$ is not Jónsson.<br /> <br /> It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.<br /> <br /> === Relations to other large cardinal notions ===<br /> <br /> Jónsson cardinals have a lot of consistency strength:<br /> * Jónsson cardinals are equiconsistent with [[Ramsey]] cardinals. &lt;cite&gt;Mitchell1997:JonssonErdosCoreModel&lt;/cite&gt;<br /> * The existence of a Jónsson cardinal $\kappa$ implies the existence of [[Zero sharp|$x^\sharp$]] for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).<br /> <br /> But in terms of size, they're (ostensibly) quite small:<br /> * A Jónsson cardinal need not be regular (assuming the consistency of a [[measurable]] cardinal).<br /> * Every Ramsey cardinal is inaccessible and Jónsson. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> * Every weakly inaccessible Jónsson is [[Mahlo|weakly hyper-Mahlo]]. &lt;cite&gt;Shelah1994:CardinalArithmetic&lt;/cite&gt;<br /> <br /> It's an open question whether or not every inaccessible Jónsson cardinal is [[weakly compact]].<br /> <br /> === Jónsson successors of singulars ===<br /> <br /> As mentioned above, $\aleph_{\omega+1}$ is not Jónsson (this is due to Shelah). The question is then if it's possible for any successor of a singular cardinal to be Jónsson. Here is a (non-exhaustive) list of things known:<br /> * If $0\neq\gamma&lt;|\eta|$ then $\aleph_{\eta+\gamma+1}$ is not Jónsson. &lt;cite&gt;TrybaJan1983:JonssonUncountable&lt;/cite&gt;<br /> * If there exists a Jónsson successor of a singular cardinal then [[Zero dagger|$0^\dagger$]] exists. &lt;cite&gt;DonderKoepke1998:AccessibleJonsson&lt;/cite&gt;<br /> <br /> == Jónsson cardinals and the core model ==<br /> <br /> In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model that can be found in &lt;cite&gt;Welch1998:InnerModels&lt;/cite&gt;. <br /> Assuming there is no inner model with a [[Woodin]] cardinal then:<br /> * Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal.<br /> * If $\kappa$ is regular Jónsson then the set of regular $\alpha&lt;\kappa$ satisfying weak covering is stationary in $\kappa$.<br /> <br /> If we assume that there's no sharp for a [[strong]] cardinal (known as $0^\P$ doesn't exist) then:<br /> * For a Jónsson cardinal $\kappa$, [[Zero sharp|$A^\sharp$]] exists for every $A\subseteq\kappa$.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Jonsson&diff=2645 Jonsson 2018-10-07T18:13:24Z <p>Zetapology: /* Embedding Characterization */</p> <hr /> <div>{{DISPLAYTITLE: Jónsson cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Partition property]]<br /> Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called '''Jónsson cardinals'''.<br /> <br /> An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of that algebra is a set $B\subseteq A$ which $B$ is closed under each $f_k$ for $k\leq n$.<br /> <br /> == Equivalent Definitions ==<br /> <br /> There are several equivalent definitions of Jónsson cardinals.<br /> <br /> === Partition Property ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff the [[partition property]] $\kappa\rightarrow [\kappa]_\kappa^{&lt;\omega}$ holds, i.e. that given any function $f:[\kappa]^{&lt;\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ of order type $\kappa$ such that $f[H]^n\neq\kappa$ for every $n&lt;\omega$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Substructure Characterization ===<br /> *A cardinal $\kappa$ is '''Jónsson''' iff given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$. <br /> <br /> *A cardinal $\kappa$ is '''Jónsson''' iff any structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Embedding Characterization ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff for every $\theta&gt;\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j&lt;\kappa$, iff for every $\theta&gt;\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to V_\theta$ such that $j(\kappa)=\kappa$.<br /> <br /> Interestingly, if one such $\theta&gt;\kappa$ has this property, then every $\theta&gt;\kappa$ has this property as well.<br /> <br /> === Abstract Algebra Characterization ===<br /> <br /> In terms of abstract algebra, $\kappa$ is '''Jónsson''' iff any algebra $A$ of size $\kappa$ has a proper subalgebra of $A$ of size $\kappa$.<br /> <br /> == Properties ==<br /> <br /> All the following facts can be found in &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;:<br /> <br /> * $\aleph_0$ is not Jónsson.<br /> * If $\kappa$ isn't Jónsson then neither is $\kappa^+$.<br /> * If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.<br /> * If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).<br /> * A singular limit of [[measurable|measurables]] is Jónsson.<br /> * The least Jónsson is either [[inaccessible|weakly inaccessible]] or has cofinality $\omega$.<br /> * $\aleph_{\omega+1}$ is not Jónsson.<br /> <br /> It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.<br /> <br /> === Relations to other large cardinal notions ===<br /> <br /> Jónsson cardinals have a lot of consistency strength:<br /> * Jónsson cardinals are equiconsistent with [[Ramsey]] cardinals. &lt;cite&gt;Mitchell1997:JonssonErdosCoreModel&lt;/cite&gt;<br /> * The existence of a Jónsson cardinal $\kappa$ implies the existence of [[Zero sharp|$x^\sharp$]] for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).<br /> <br /> But in terms of size, they're (ostensibly) quite small:<br /> * A Jónsson cardinal need not be regular (assuming the consistency of a [[measurable]] cardinal).<br /> * Every Ramsey cardinal is inaccessible and Jónsson. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> * Every weakly inaccessible Jónsson is [[Mahlo|weakly hyper-Mahlo]]. &lt;cite&gt;Shelah1994:CardinalArithmetic&lt;/cite&gt;<br /> <br /> It's an open question whether or not every inaccessible Jónsson cardinal is [[weakly compact]].<br /> <br /> === Jónsson successors of singulars ===<br /> <br /> As mentioned above, $\aleph_{\omega+1}$ is not Jónsson (this is due to Shelah). The question is then if it's possible for any successor of a singular cardinal to be Jónsson. Here is a (non-exhaustive) list of things known:<br /> * If $0\neq\gamma&lt;|\eta|$ then $\aleph_{\eta+\gamma+1}$ is not Jónsson. &lt;cite&gt;TrybaJan1983:JonssonUncountable&lt;/cite&gt;<br /> * If there exists a Jónsson successor of a singular cardinal then [[Zero dagger|$0^\dagger$]] exists. &lt;cite&gt;DonderKoepke1998:AccessibleJonsson&lt;/cite&gt;<br /> <br /> == Jónsson cardinals and the core model ==<br /> <br /> In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model that can be found in &lt;cite&gt;Welch1998:InnerModels&lt;/cite&gt;. <br /> Assuming there is no inner model with a [[Woodin]] cardinal then:<br /> * Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal.<br /> * If $\kappa$ is regular Jónsson then the set of regular $\alpha&lt;\kappa$ satisfying weak covering is stationary in $\kappa$.<br /> <br /> If we assume that there's no sharp for a [[strong]] cardinal (known as $0^\P$ doesn't exist) then:<br /> * For a Jónsson cardinal $\kappa$, [[Zero sharp|$A^\sharp$]] exists for every $A\subseteq\kappa$.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Jonsson&diff=2644 Jonsson 2018-10-07T18:10:55Z <p>Zetapology: /* Equivalent Definitions */ added abstract algebra characterization</p> <hr /> <div>{{DISPLAYTITLE: Jónsson cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Partition property]]<br /> Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called '''Jónsson cardinals'''.<br /> <br /> An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of that algebra is a set $B\subseteq A$ which $B$ is closed under each $f_k$ for $k\leq n$.<br /> <br /> == Equivalent Definitions ==<br /> <br /> There are several equivalent definitions of Jónsson cardinals.<br /> <br /> === Partition Property ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff the [[partition property]] $\kappa\rightarrow [\kappa]_\kappa^{&lt;\omega}$ holds, i.e. that given any function $f:[\kappa]^{&lt;\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ of order type $\kappa$ such that $f[H]^n\neq\kappa$ for every $n&lt;\omega$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Substructure Characterization ===<br /> *A cardinal $\kappa$ is '''Jónsson''' iff given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$. <br /> <br /> *A cardinal $\kappa$ is '''Jónsson''' iff any structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;<br /> <br /> === Embedding Characterization ===<br /> A cardinal $\kappa$ is '''Jónsson''' iff for every $\theta&gt;\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j&lt;\kappa$.<br /> <br /> Interestingly, if one such $\theta&gt;\kappa$ has this property, then every $\theta&gt;\kappa$ has this property as well.<br /> <br /> === Abstract Algebra Characterization ===<br /> <br /> In terms of abstract algebra, $\kappa$ is '''Jónsson''' iff any algebra $A$ of size $\kappa$ has a proper subalgebra of $A$ of size $\kappa$.<br /> <br /> == Properties ==<br /> <br /> All the following facts can be found in &lt;cite&gt;Kanamori2003:HigherInfinite&lt;/cite&gt;:<br /> <br /> * $\aleph_0$ is not Jónsson.<br /> * If $\kappa$ isn't Jónsson then neither is $\kappa^+$.<br /> * If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.<br /> * If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).<br /> * A singular limit of [[measurable|measurables]] is Jónsson.<br /> * The least Jónsson is either [[inaccessible|weakly inaccessible]] or has cofinality $\omega$.<br /> * $\aleph_{\omega+1}$ is not Jónsson.<br /> <br /> It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.<br /> <br /> === Relations to other large cardinal notions ===<br /> <br /> Jónsson cardinals have a lot of consistency strength:<br /> * Jónsson cardinals are equiconsistent with [[Ramsey]] cardinals. &lt;cite&gt;Mitchell1997:JonssonErdosCoreModel&lt;/cite&gt;<br /> * The existence of a Jónsson cardinal $\kappa$ implies the existence of [[Zero sharp|$x^\sharp$]] for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).<br /> <br /> But in terms of size, they're (ostensibly) quite small:<br /> * A Jónsson cardinal need not be regular (assuming the consistency of a [[measurable]] cardinal).<br /> * Every Ramsey cardinal is inaccessible and Jónsson. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> * Every weakly inaccessible Jónsson is [[Mahlo|weakly hyper-Mahlo]]. &lt;cite&gt;Shelah1994:CardinalArithmetic&lt;/cite&gt;<br /> <br /> It's an open question whether or not every inaccessible Jónsson cardinal is [[weakly compact]].<br /> <br /> === Jónsson successors of singulars ===<br /> <br /> As mentioned above, $\aleph_{\omega+1}$ is not Jónsson (this is due to Shelah). The question is then if it's possible for any successor of a singular cardinal to be Jónsson. Here is a (non-exhaustive) list of things known:<br /> * If $0\neq\gamma&lt;|\eta|$ then $\aleph_{\eta+\gamma+1}$ is not Jónsson. &lt;cite&gt;TrybaJan1983:JonssonUncountable&lt;/cite&gt;<br /> * If there exists a Jónsson successor of a singular cardinal then [[Zero dagger|$0^\dagger$]] exists. &lt;cite&gt;DonderKoepke1998:AccessibleJonsson&lt;/cite&gt;<br /> <br /> == Jónsson cardinals and the core model ==<br /> <br /> In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model that can be found in &lt;cite&gt;Welch1998:InnerModels&lt;/cite&gt;. <br /> Assuming there is no inner model with a [[Woodin]] cardinal then:<br /> * Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal.<br /> * If $\kappa$ is regular Jónsson then the set of regular $\alpha&lt;\kappa$ satisfying weak covering is stationary in $\kappa$.<br /> <br /> If we assume that there's no sharp for a [[strong]] cardinal (known as $0^\P$ doesn't exist) then:<br /> * For a Jónsson cardinal $\kappa$, [[Zero sharp|$A^\sharp$]] exists for every $A\subseteq\kappa$.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=N-fold_variants&diff=2643 N-fold variants 2018-10-07T17:20:11Z <p>Zetapology: /* $\omega$-fold strong */ minor correction</p> <hr /> <div>{{DISPLAYTITLE: $n$-fold Variants of Large Cardinals}}<br /> [[Category: Large cardinal axioms]]<br /> [[Category: Critical points]]<br /> ''This page is a WIP.''<br /> The $n$-fold variants of large cardinal axioms were created by Sato Kentaro in &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt; in order to study and investigate the double helix phenomena. The double helix phenomena is the strange pattern in consistency strength between such cardinals, which can be seen below.<br /> <br /> [[file:DoubleHelix.png | 600px | center]]<br /> <br /> This diagram was created by Kentaro. The arrows denote consistency strength, and the double lines denote equivalency. The large cardinals in this diagram will be detailed on this page (unless found elsewhere on this website).<br /> <br /> This page will only use facts from &lt;cite&gt;Kentaro2007:DoubleHelix&lt;/cite&gt; unless otherwise stated.<br /> <br /> == $n$-fold Variants ==<br /> <br /> The $n$-fold variants of large cardinals were given in a very large paper by Sato Kentaro. Most of the definitions involve giving large closure properties to the $M$ used in the original large cardinal in an [[elementary embedding]] $j:V\rightarrow M$. They are very large, but [[rank into rank | rank-into-rank]] cardinals are stronger than most $n$-fold variants of large cardinals. <br /> <br /> Generally, the $n$-fold variant of a large cardinal axiom is the similar to the generalization of [[superstrong]] cardinals to [[superstrong|$n$-superstrong]] cardinals, [[huge]] cardinals to [[huge|$n$-huge]] cardinals, etc. More specifically, if the definition of the original axiom is that $j:V\prec M$ has critical point $\kappa$ and $M$ has some closure property which uses $\kappa$, then the definition of the $n$-fold variant of the axiom is that $M$ has that closure property on $j^n{\kappa}$.<br /> <br /> === $n$-fold Variants Which Are Simply the Original Large Cardinal ===<br /> <br /> There were many $n$-fold variants which were simply different names of the original large cardinal. This was due to the fact that some n-fold variants, if only named n-variants instead, would be confusing to the reader (for example the $n$-fold extendibles rather than the [[extendible | $n$-extendibles]]). Here are a list of such cardinals:<br /> <br /> *The '''$n$-fold superstrong''' cardinals are precisely the [[superstrong | $n$-superstrong]] cardinals<br /> *The '''$n$-fold almost huge''' cardinals are precisely the [[huge | almost $n$-huge]] cardinals<br /> *The '''$n$-fold huge''' cardinals are precisely the [[huge | $n$-huge]] cardinals<br /> *The '''$n$-fold superhuge''' cardinals are precisely the [[huge | $n$-superhuge]] cardinals<br /> *The '''$\omega$-fold superstrong''' and '''$\omega$-fold Shelah''' cardinals are precisely the [[rank-into-rank|I2]] cardinals<br /> <br /> === $n$-fold supercompact cardinals ===<br /> <br /> A cardinal $\kappa$ is '''$n$-fold $\lambda$-supercompact''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\lambda&lt;j(\kappa)$ and $M^{j^{n-1}(\lambda)}\subset M$ (i.e. $M$ is closed under all of its sequences of length $j^{n-1}(\lambda)$). This definition is very similar to that of the [[huge | $n$-huge]] cardinals.<br /> <br /> A cardinal $\kappa$ is '''$n$-fold supercompact''' iff it is $n$-fold $\lambda$-supercompact for every $\lambda$. Consistency-wise, the $n$-fold supercompact cardinals are stronger than the [[superstrong | $n$-superstrong]] cardinals and weaker than the $(n+1)$-fold strong cardinals. In fact, if an $n$-fold supercompact cardinal exists, then it is consistent for there to be a proper class of $n$-superstrong cardinals.<br /> <br /> It is clear that the $n+1$-fold $0$-supercompact cardinals are precisely the [[huge|$n$-huge]] cardinals. The $1$-fold supercompact cardinals are precisely the [[supercompact]] cardinals. The $0$-fold supercompact cardinals are precisely the [[measurable]] cardinals.<br /> <br /> === $n$-fold strong cardinals ===<br /> <br /> A cardinal $\kappa$ is '''$n$-fold $\lambda$-strong''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\kappa+\lambda&lt;j(\kappa)$ and $V_{j^{n-1}(\kappa+\lambda)}\subset M$.<br /> <br /> A cardinal $\kappa$ is '''$n$-fold strong''' iff it is $n$-fold $\lambda$-strong for every $\lambda$. Consistency-wise, the $(n+1)$-fold strong cardinals are stronger than the $n$-fold supercompact cardinals, equivalent to the $n$-fold extendible cardinals, and weaker than the $(n+1)$-fold Woodin cardinals. More specifically, in the rank of an (n+1)-fold Woodin cardinal there is an $(n+1)$-fold strong cardinal.<br /> <br /> It is clear that the $(n+1)$-fold $0$-strong cardinals are precisely the [[superstrong|$n$-superstrong]] cardinals. The $1$-fold strong cardinals are precisely the [[strong]] cardinals. The $0$-fold strong cardinals are precisely the [[measurable]] cardinals.<br /> <br /> === $n$-fold extendible cardinals ===<br /> <br /> ''(To be added)''<br /> <br /> === $n$-fold Woodin cardinals ===<br /> <br /> A cardinal $\kappa$ is '''$n$-fold Woodin''' iff for every function $f:\kappa\rightarrow\kappa$ there is some ordinal $\alpha&lt;\kappa$ such that $\{f(\beta):\beta&lt;\alpha\}\subseteq\alpha$ and $V_{j^{n}(f)(j^{n-1}(\alpha))}\subset M$. Consistency-wise, the $(n+1)$-fold Woodin cardinals are stronger than the $(n+1)$-fold strong cardinals, and weaker than the $(n+1)$-fold Shelah cardinals. Specifically, in the rank of an $(n+1)$-fold Shelah cardinal there is an $(n+1)$-fold Woodin cardinal, and every $(n+1)$-fold Shelah cardinal is also an $(n+1)$-fold Woodin cardinal.<br /> <br /> The $2$-fold Woodin cardinals are precisely the [[Vopenka|Vopěnka]] cardinals (therefore precisely the [[Woodin#Shelah|Woodin for supercompactness]] cardinals). In fact, the $n+1$-fold Woodin cardinals are precisely the $n$-fold Vopěnka cardinals. The $1$-fold Woodin cardinals are precisely the [[Woodin]] cardinals.<br /> <br /> ''(More to be added)''<br /> <br /> == $\omega$-fold variants ==<br /> <br /> The $\omega$-fold variant is a very strong version of the $n$-fold variant, to the point where they even beat some of the [[rank-into-rank]] axioms in consistency strength. Interestingly, they follow a somewhat backwards pattern of consistency strength relative to the original double helix. For example, $n$-fold strong is much weaker than $n$-fold Vopěnka (the jump is similar to the jump between a [[strong]] cardinal and a [[Vopenka|Vopěnka]] cardinal), but $\omega$-fold strong is much, much stronger than $\omega$-fold Vopěnka.<br /> <br /> === $\omega$-fold extendible ===<br /> <br /> ''(To be added)''<br /> <br /> === $\omega$-fold Vopěnka ===<br /> <br /> ''(To be added)''<br /> <br /> === $\omega$-fold Woodin ===<br /> <br /> A cardinal $\kappa$ is '''$\omega$-fold Woodin''' iff for every function $f:\kappa\rightarrow\kappa$ there is some ordinal $\alpha&lt;\kappa$ such that $\{f(\beta):\beta&lt;\alpha\}\subseteq\alpha$ and $V_{j^{\omega}(f)(\alpha))}\subset M$.<br /> <br /> Consistency-wise, the existence of an $\omega$-fold Woodin cardinal is stronger than the [[rank-into-rank|I2]] axiom, but weaker than the existence of an $\omega$-fold strong cardinal. In particular, if there is an $\omega$-fold strong cardinal $\kappa$ then $\kappa$ is $\omega$-fold Woodin and has $\kappa$-many $\omega$-fold Woodin cardinals below it, and $V_\kappa$ satisfies the existence of a proper class of $\omega$-fold Woodin cardinals.<br /> <br /> === $\omega$-fold strong ===<br /> <br /> A cardinal $\kappa$ is '''$\omega$-fold $\lambda$-strong''' iff it is the critical point of some nontrivial elementary embedding $j:V\rightarrow M$ such that $\kappa+\lambda&lt;j(\kappa)$ and $V_{j^\omega(\kappa+\lambda)}\subset M$.<br /> <br /> $\kappa$ is '''$\omega$-fold strong''' iff it is $\omega$-fold $\lambda$-strong for every $\lambda$.<br /> <br /> Consistency-wise, the existence of an $\omega$-fold strong cardinal is stronger than the existence of an $\omega$-fold Woodin cardinal and weaker than the assertion that there is a $\Sigma_4^1$-elementary embedding $j:V_\lambda\prec V_\lambda$ with an uncountable critical point $\kappa&lt;\lambda$ (this is a weakening of the [[rank-into-rank|I1]] axiom known as $E_2$). In particular, if there is a cardinal $\kappa$ which is the critical point of some elementary embedding witnessing the $E_2$ axiom, then there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]] over $\kappa$ which contains the set of all cardinals which are $\omega$-fold strong in $V_\kappa$ and therefore $V_\kappa$ satisfies the existence of a proper class of $\omega$-fold strong cardinals.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Strong&diff=2642 Strong 2018-10-07T16:55:43Z <p>Zetapology: /* Elementary Embedding Characterization */</p> <hr /> <div>{{DISPLAYTITLE: Strong cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> Strong cardinals were created as a weakening of [[supercompact]] cardinals introduced by Dodd and Jensen in 1982 &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;. They are defined as a strengthening of [[measurable|measurability]], being that they are critical points of [[elementary embedding|elementary embeddings]] $j:V\rightarrow M$ for some transitive inner model of [[ZFC]] $M$. Hypermeasurability is a weakening of strongness (the property of being a strong cardinal is often called strongness), although if $\lambda=\beth_\lambda$ then a cardinal is $\lambda$-strong iff it is $\lambda$-hypermeasurable.<br /> <br /> == Definitions of Strongness ==<br /> <br /> There are multiple equivalent definitions of strongness, using [[elementary embedding|elementary embeddings]] and [[extender|extenders]].<br /> <br /> === Elementary Embedding Characterization ===<br /> <br /> A cardinal $\kappa$ is '''$\gamma$-strong''' iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $V_\gamma\subset M$. A cardinal $\kappa$ is '''strong''' iff it is $\gamma$-strong for each $\gamma$, iff it is $\gamma$-strong for arbitrarily large $\gamma$, iff for each set $x$, $\kappa$ is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$.<br /> <br /> More intuitively, there are elementary embeddings from $V$ into transitive classes which have critical point $\kappa$ and (in total) contain any set one wishes.<br /> <br /> === Extender Characterization ===<br /> <br /> A cardinal $\kappa$ is '''strong''' iff it is [[uncountable]] and for every set $X$ of rank $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-extender $E$ such that, letting the [[ultrapower]] of $V$ by $E$ be called $Ult_E$ and the canonical ultrapower embedding from $V$ to $Ult_E$ be called $j$, $X\in Ult_E$ and $\lambda&lt;j(\kappa)$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.<br /> <br /> == Definitions of Hypermeasurability ==<br /> <br /> The definitions of hypermeasurability are very similar to the definitions of strongness, mainly because hypermeasurability is a generalized version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness.<br /> <br /> === Elementary Embedding Characterization ===<br /> <br /> A cardinal $\kappa$ is '''$x$-hypermeasurable''' for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is '''$\lambda$-hypermeasurable''' iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of [[Hereditary Cardinality|hereditary cardinality]] less than $\lambda$).<br /> <br /> Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$.<br /> <br /> == Facts about Strongness and Hypermeasurability ==<br /> <br /> Here is a list of facts about these cardinals:<br /> <br /> *A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.<br /> *In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.<br /> *A cardinal $\kappa$ is [[measurable]] if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.<br /> *If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; <br /> *Every [[Shelah]] cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda&lt;\kappa:\lambda$ is strong$\}$ <br /> *The [[Mitchell rank]] of any strong cardinal $o(\kappa)=(2^\kappa)^+$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; <br /> *Any strong cardinal is [[reflecting|$\Sigma_2$-reflecting]]. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> *Every strong cardinal is [[unfoldable|strongly unfoldable]] and thus [[indescribable|totally indescribable]]. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt; Therefore, each of the following is never strong:<br /> **The least [[measurable]] cardinal.<br /> **The least $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]], the least $\kappa$ which is [[supercompact|$2^{2^\kappa}$-supercompact]], etc.<br /> **For each $n$, the least [[huge|$n$-huge]] index cardinal (that is, the least ''target'' of an embedding witnessing $n$-hugeness of some cardinal) and the least $n$-huge cardinal.<br /> **For each $n&lt;\omega$, The least $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda&lt;\kappa$ (see [[extendible|$n$-extendible]]).<br /> **The least $\kappa$ which is both $2^\kappa$-supercompact and [[Vopenka|Vopěnka]], the least $\kappa$ which is both $2^{2^\kappa}$-supercompact and Vopěnka, etc., the least $\kappa$ which is both measurable and Vopěnka, for each $n$ the least Vopěnka $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda&lt;\kappa$, and more.<br /> *If there is a strong cardinal then $V\neq L[A]$ for every set $A$.<br /> *Assuming both a strong cardinal and a [[superstrong]] cardinal exist, and the least strong cardinal $\kappa$ has a superstrong above it, then the least strong cardinal has $\kappa$ superstrong cardinals below it.<br /> <br /> == Core Model up to Strongness ==<br /> <br /> Dodd and Jensen created a [[core model]] based on sequences of [[extender|extenders]] of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> *[[L|$L[\mathcal{E}]$]] is an inner model of [[ZFC]].<br /> *$L[\mathcal{E}]$ satisfies [[GCH]], the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.<br /> *$L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal<br /> *If there does not exist an inner model of the existence of a strong cardinal, then:<br /> **There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$<br /> **If $\kappa$ is a singular [[Beth|strong limit]] cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$ <br /> <br /> As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a &quot;sharp&quot; defined analogously to [[Zero sharp|$0^{\#}$]], but instead of using $L$ one uses $L[\mathcal{E}]$. They then showed that there is a nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$ iff such a real exists. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; This real is commonly referred to as ''the sharp for a strong cardinal''.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2641 Tall 2018-10-07T16:35:13Z <p>Zetapology: /* Extender Characterization */ Fixed some characterizations, added tall extender characterization</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta$-tall iff there exists some $(\kappa,\theta^+)$-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 $(\kappa,\lambda)$-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 $j(\kappa)&gt;\theta$. $\kappa$ is '''strongly tall''' iff it is strongly $\theta$-tall for every $\theta$. It is not known whether or not all [[strong]] cardinals are strongly tall, although every [[strongly compact|strongly $\theta$-compact]] cardinal is strongly $\theta$-tall. It is conjectured that strongly tall cardinals are equiconsistent with strong cardinals (and therefore with tall cardinals).<br /> <br /> === Ultrapower Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and an [[filter|ultrafilter]] $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<br /> <br /> === Ultrafilter Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff there is some set $S$, an [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:<br /> #$\kappa$ is uncountable.<br /> #$U$ has completeness $\kappa^+$.<br /> #$H_0(x)=0$ for each $x\in S$.<br /> #For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=\beta$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2640 Tall 2018-10-07T06:58:28Z <p>Zetapology: /* Tall Cardinals */</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<br /> <br /> === Extender Characterization ===<br /> <br /> If $\theta$ is a cardinal, $\kappa$ is $\theta^+$-tall iff there exists some $(\kappa,\theta)$-extender $E$ such that, if $j:V\prec M\cong Ult_E$ is the ultrapower embedding of $V$ by $E$, $M^\kappa\subset M$.<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 $j(\kappa)&gt;\theta$. $\kappa$ is '''strongly tall''' iff it is strongly $\theta$-tall for every $\theta$. It is not known whether or not all [[strong]] cardinals are strongly tall, although every [[strongly compact|strongly $\theta$-compact]] cardinal is strongly $\theta$-tall. It is conjectured that strongly tall cardinals are equiconsistent with strong cardinals (and therefore with tall cardinals).<br /> <br /> === Ultrapower Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and an [[filter|ultrafilter]] $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<br /> <br /> === Ultrafilter Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff there is some set $S$, an [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:<br /> #$\kappa$ is uncountable.<br /> #$U$ has completeness $\kappa^+$.<br /> #$H_0(x)=0$ for each $x\in S$.<br /> #For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=\beta$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Tall&diff=2639 Tall 2018-10-07T06:33:02Z <p>Zetapology: added strongly tall cardinals</p> <hr /> <div>{{DISPLAYTITLE: Tall cardinal}}<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 $j(\kappa)&gt;\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ 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 &lt;cite&gt;Hamkins2009:TallCardinals&lt;/cite&gt;.<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 $j(\kappa)&gt;\theta$. $\kappa$ is '''strongly tall''' iff it is strongly $\theta$-tall for every $\theta$. It is not known whether or not all [[strong]] cardinals are strongly tall, although every [[strongly compact|strongly $\theta$-compact]] cardinal is strongly $\theta$-tall. It is conjectured that strongly tall cardinals are equiconsistent with strong cardinals (and therefore with tall cardinals).<br /> <br /> === Ultrapower Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and an [[filter|ultrafilter]] $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)&gt;\theta$.<br /> <br /> === Ultrafilter Characterization ===<br /> <br /> $\kappa$ is strongly $\theta$-tall iff there is some set $S$, an [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:<br /> #$\kappa$ is uncountable.<br /> #$U$ has completeness $\kappa^+$.<br /> #$H_0(x)=0$ for each $x\in S$.<br /> #For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)&lt;H_\alpha(x)\}\in U$ iff there is some $\beta&lt;\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)&lt;H_\alpha(x)$ almost everywhere iff there is some $\beta&lt;\alpha$ such that $f(x)=\beta$ almost everywhere.<br /> #$\{x\in S:H_\theta(x)&lt;\kappa\}\in U$. That is, $H_\theta(x)&lt;\kappa$ almost everywhere.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Upper_attic&diff=2638 Upper attic 2018-10-07T05:58:37Z <p>Zetapology: added strongly tall cardinals</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 /> * The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal<br /> * '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings<br /> * The [[wholeness axioms]]<br /> * [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...<br /> * '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[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 cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal<br /> * [[Vopenka | Vopěnka's principle]]<br /> * [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal<br /> &lt;!--* [[grand reflection]] cardinal--&gt;<br /> * [[hypercompact]] cardinal<br /> * '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal<br /> * '''[[strongly compact]]''' cardinal [[strongly compact | $\lambda$-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<br /> * [[Woodin#Shelah|Shelah]] cardinal<br /> * The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br /> * '''[[Woodin]]''' cardinal<br /> * [[tall|strongly tall]] cardinals (possibly equiconsistent to strong cardinals)<br /> * [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy<br /> * Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br /> *[[zero dagger| $0^\dagger$]] (''zero-dagger'')<br /> * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal<br /> * [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal<br /> * [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal<br /> * '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] 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<br /> * '''[[zero sharp | $0^\sharp$]]''' (''zero-sharp''), existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]<br /> * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br /> * the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq\alpha&lt;\omega_1$<br /> * [[remarkable]] cardinal<br /> * [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal<br /> * [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal<br /> * [[subtle]] cardinal<br /> * [[ineffable#Ethereal cardinal|ethereal]] cardinal<br /> * [[unfoldable#Superstrongly Unfoldable | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br /> * [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br /> * [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable | strongly unfoldable]] cardinal<br /> * [[indescribable]] hierarchy, [[totally indescribable]] cardinal<br /> * '''[[weakly compact]]''' cardinal<br /> * The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$ <br /> * '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals<br /> * [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal<br /> * [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<br /> * [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] 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 and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals<br /> * [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals<br /> * '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly 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 | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]<br /> * '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br /> <br /> * down to [[the middle attic]]</div> Zetapology http://cantorsattic.info/index.php?title=Strong&diff=2637 Strong 2018-10-06T04:13:06Z <p>Zetapology: /* Extender Characterization */</p> <hr /> <div>{{DISPLAYTITLE: Strong cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> Strong cardinals were created as a weakening of [[supercompact]] cardinals introduced by Dodd and Jensen in 1982 &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;. They are defined as a strengthening of [[measurable|measurability]], being that they are critical points of [[elementary embedding|elementary embeddings]] $j:V\rightarrow M$ for some transitive inner model of [[ZFC]] $M$. Hypermeasurability is a weakening of strongness (the property of being a strong cardinal is often called strongness), although if $\lambda=\beth_\lambda$ then a cardinal is $\lambda$-strong iff it is $\lambda$-hypermeasurable.<br /> <br /> == Definitions of Strongness ==<br /> <br /> There are multiple equivalent definitions of strongness, using [[elementary embedding|elementary embeddings]] and [[extender|extenders]].<br /> <br /> === Elementary Embedding Characterization ===<br /> <br /> A cardinal $\kappa$ is '''$\gamma$-strong''' iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $V_\gamma\subset M$. A cardinal $\kappa$ is '''strong''' iff it is $\gamma$-strong for each $\gamma$, iff it is $\gamma$-strong for arbitrarily large $\gamma$, iff for each set $x$, $\kappa$ is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$.<br /> <br /> More intuitively, there are elementary embeddings from $V$ into transitive classes which have critical point $\kappa$ and contain arbitrarily large initial segments of the universe.<br /> <br /> === Extender Characterization ===<br /> <br /> A cardinal $\kappa$ is '''strong''' iff it is [[uncountable]] and for every set $X$ of rank $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-extender $E$ such that, letting the [[ultrapower]] of $V$ by $E$ be called $Ult_E$ and the canonical ultrapower embedding from $V$ to $Ult_E$ be called $j$, $X\in Ult_E$ and $\lambda&lt;j(\kappa)$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.<br /> <br /> == Definitions of Hypermeasurability ==<br /> <br /> The definitions of hypermeasurability are very similar to the definitions of strongness, mainly because hypermeasurability is a generalized version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness.<br /> <br /> === Elementary Embedding Characterization ===<br /> <br /> A cardinal $\kappa$ is '''$x$-hypermeasurable''' for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is '''$\lambda$-hypermeasurable''' iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of [[Hereditary Cardinality|hereditary cardinality]] less than $\lambda$).<br /> <br /> Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$.<br /> <br /> == Facts about Strongness and Hypermeasurability ==<br /> <br /> Here is a list of facts about these cardinals:<br /> <br /> *A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.<br /> *In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.<br /> *A cardinal $\kappa$ is [[measurable]] if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.<br /> *If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; <br /> *Every [[Shelah]] cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda&lt;\kappa:\lambda$ is strong$\}$ <br /> *The [[Mitchell rank]] of any strong cardinal $o(\kappa)=(2^\kappa)^+$. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; <br /> *Any strong cardinal is [[reflecting|$\Sigma_2$-reflecting]]. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> *Every strong cardinal is [[unfoldable|strongly unfoldable]] and thus [[indescribable|totally indescribable]]. &lt;cite&gt;Gitman2011:RamseyLikeCardinals&lt;/cite&gt; Therefore, each of the following is never strong:<br /> **The least [[measurable]] cardinal.<br /> **The least $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]], the least $\kappa$ which is [[supercompact|$2^{2^\kappa}$-supercompact]], etc.<br /> **For each $n$, the least [[huge|$n$-huge]] index cardinal (that is, the least ''target'' of an embedding witnessing $n$-hugeness of some cardinal) and the least $n$-huge cardinal.<br /> **For each $n&lt;\omega$, The least $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda&lt;\kappa$ (see [[extendible|$n$-extendible]]).<br /> **The least $\kappa$ which is both $2^\kappa$-supercompact and [[Vopenka|Vopěnka]], the least $\kappa$ which is both $2^{2^\kappa}$-supercompact and Vopěnka, etc., the least $\kappa$ which is both measurable and Vopěnka, for each $n$ the least Vopěnka $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda&lt;\kappa$, and more.<br /> *If there is a strong cardinal then $V\neq L[A]$ for every set $A$.<br /> *Assuming both a strong cardinal and a [[superstrong]] cardinal exist, and the least strong cardinal $\kappa$ has a superstrong above it, then the least strong cardinal has $\kappa$ superstrong cardinals below it.<br /> <br /> == Core Model up to Strongness ==<br /> <br /> Dodd and Jensen created a [[core model]] based on sequences of [[extender|extenders]] of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> *[[L|$L[\mathcal{E}]$]] is an inner model of [[ZFC]].<br /> *$L[\mathcal{E}]$ satisfies [[GCH]], the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.<br /> *$L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal<br /> *If there does not exist an inner model of the existence of a strong cardinal, then:<br /> **There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$<br /> **If $\kappa$ is a singular [[Beth|strong limit]] cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$ <br /> <br /> As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a &quot;sharp&quot; defined analogously to [[Zero sharp|$0^{\#}$]], but instead of using $L$ one uses $L[\mathcal{E}]$. They then showed that there is a nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$ iff such a real exists. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; This real is commonly referred to as ''the sharp for a strong cardinal''.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Strongly_compact&diff=2636 Strongly compact 2018-09-29T19:13:08Z <p>Zetapology: /* Relation to other large cardinal notions */ relation to tall cardinals</p> <hr /> <div>{{DISPLAYTITLE: Strongly compact cardinal}}<br /> [[Category:Large cardinal axioms]]<br /> [[Category:Critical points]]<br /> The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for an<br /> uncountable cardinal $\kappa$ is ''strongly compact'' if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It turns out that this model-theoretic concept admits fruitful embedding characterizations, which as with so many large cardinal notions, has become the focus of study. Strong compactness rarefies into a hierarchy, and a cardinal $\kappa$ is strongly compact if and only if it is $\theta$-strongly compact for every ordinal $\theta\geq\kappa$. <br /> <br /> The strongly compact embedding characterizations are closely related to that of [[supercompact]] cardinals, which are characterized by [[elementary embedding|elementary embeddings]] with a high degree of closure: $\kappa$ is $\theta$-[[supercompact]] if and only if there is an embedding $j:V\to M$ with critical point $\kappa$ such that $\theta&lt;j(\kappa)$ and every subset of $M$ of size $\theta$ is an element of $M$. By weakening this closure requirement to insist only that $M$ contains a small cover for any subset of size $\theta$, or even just a small cover of the set $j''\theta$ itself, we arrive at the $\theta$-strongly compact cardinals. It follows that every $\theta$-[[supercompact]] cardinal is $\theta$-strongly compact and so every supercompact cardinal is strongly compact. Furthermore, since every ultrapower embedding $j:V\to M$ with critical point $\kappa$ has $M^\kappa\subset M$, for $\theta$-strong compactness we may restrict our attention to the case when $\kappa\leq\theta$.<br /> <br /> == Diverse characterizations ==<br /> <br /> There are diverse equivalent characterizations of the strongly compact cardinals. <br /> <br /> === Strong compactness characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is ''strongly compact'' if every $\kappa$-satisfiable theory in the infinitary logic $L_{\kappa,\kappa}$ is satisfiable. The signature of an $L_{\kappa,\kappa}$ language consists, just as in the first order context, of a set of finitary function, relation and constant symbols. The $L_{\kappa,\kappa}$ formulas, however, are built up in an infinitary process, by closing under infinitary conjunctions $\wedge_{\alpha&lt;\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha&lt;\delta}\varphi_\alpha$ of any size $\delta&lt;\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\langle x_\alpha\mid\alpha&lt;\delta\rangle$ of size less than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics. A theory is ''$\kappa$-satisfiable'' if every subtheory consisting of fewer than $\kappa$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical compactness theorem asserts that every $\omega$-satisfiable $L_{\omega,\omega}$ theory is satisfiable. Similarly, an uncountable cardinal $\kappa$ is defined to be ''strongly compact'' if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable (and we call this the ''$\kappa$-compactness property}''). The cardinal $\kappa$ is [[weakly compact]], in contrast, if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.<br /> <br /> === Strong compactness embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some set $s\in M$ with $|s|^M\lt j(\kappa)$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Cover property characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$, with critical point $\kappa$, that exhibits the ''$\theta$-strong compactness cover property'', meaning that for every $t\subset M$ of size $\theta$ there is $s\in M$ with $t\subset s$ and $|s|^M&lt;j(\kappa)$.<br /> <br /> === Fine measure characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a [[filter|fine measure]] on $\mathcal{P}_\kappa(\theta)$. The notation $\mathcal{P}_\kappa(\theta)$ means $\{\sigma\subset\theta\mid |\sigma|&lt;\kappa\}$. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Filter extension characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if every $\kappa$-complete [[filter]] of size at most $\theta$ on a set extends to a $\kappa$-complete ultrafilter on that set. &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;<br /> <br /> === Discontinuous ultrapower characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an ultrapower embedding $j:V\to M$ with critical point $\kappa$, such that $\sup j''\lambda&lt;j(\lambda)$ for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. In other words, the embedding is discontinuous at all such $\lambda$. <br /> <br /> === Discontinuous embedding characterization ===<br /> <br /> A cardinal $\kappa$ is $\theta$-strongly compact if and only if for every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$, there is an embedding $j:V\to M$ with critical point $\kappa$ and $\sup j''\lambda&lt;j(\lambda)$.<br /> <br /> === Ketonen characterization ===<br /> <br /> An uncountable regular cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $\kappa$-complete uniform ultrafilter on every regular $\lambda$ with $\kappa\leq\lambda\leq\theta^{\lt\kappa}$. An ultrafilter $\mu$ on a cardinal $\lambda$ is ''uniform'' if all final segments $[\beta,\lambda)= \{\alpha&lt;\lambda\mid \beta\leq\alpha\}$ are in $\mu$. When $\lambda$ is regular, this is equivalent to requiring that all elements of $\mu$ have the same cardinality. <br /> <br /> === Regular ultrafilter characterization ===<br /> <br /> An uncountable cardinal $\kappa$ is $\theta$-strongly compact if and only if there is a $(\kappa,\theta)$-regular ultrafilter on some set. An ultrafilter $\mu$ is ''$(\kappa,\theta)$-regular'' if it is $\kappa$-complete and there is a family $\{X_\alpha\mid\alpha&lt;\theta\}\subset \mu$ such that $\bigcap_{\alpha\in I}X_\alpha=\emptyset$ for any $I$ with $|I|=\kappa$.<br /> <br /> == Strongly compact cardinals and forcing ==<br /> <br /> If there is proper class-many strongly compact cardinals, then there is a [[forcing|generic model]] of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot;. If each strongly compact cardinal is a limit of measurable cardinals, and if the limit of any sequence of strongly compact cardinals is singular, then there is a forcing extension V[G] that is a symmetric model of $\text{ZF}$ + &quot;all uncountable cardinals are singular&quot; + &quot;every uncountable cardinal is both almost [[Ramsey]] and a [[Rowbottom]] cardinal carrying a Rowbottom filter&quot;. <br /> This also directly follows from the existence of a proper class of supercompact cardinals, as every supercomact cardinal is simultaneously strongly compact and a limit of measurable cardinals.<br /> <br /> == Relation to other large cardinal notions ==<br /> <br /> Strongly compact cardinals are [[measurable]]. The least strongly compact cardinal can be equal to the least measurable cardinal, or to the least [[supercompact]] cardinal, by results of Magidor. &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt; (It cannot be equal to both at once because the least measurable cardinal cannot be supercompact.)<br /> <br /> Even though strongly compact cardinals imply the consistency of the negation of the singular cardinal hypothesis, SCH, for any singular strong limit cardinal $\kappa$ above the least strongly compact cardinal, $2^\kappa=\kappa^+$ (also known as &quot;SCH holds above strong compactness&quot;). &lt;cite&gt;Jech2003:SetTheory&lt;/cite&gt;<br /> <br /> If there is a strongly compact cardinal $\kappa$ then for all $\lambda\geq\kappa$ and $A\subseteq\lambda$, $\lambda^+$ is [[ineffable]] in $L[A]$.<br /> <br /> 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.<br /> <br /> Every strongly compact cardinal is [[tall]], although the existence of a strongly compact cardinal is equiconsistent with &quot;the least measurable cardinal is the least strongly compact cardinal, and therefore the least tall cardinal&quot; meaning strongly compact cardinals aren't necessarily limits of tall cardinals.<br /> <br /> {{References}}</div> Zetapology http://cantorsattic.info/index.php?title=Kunen_inconsistency&diff=2635 Kunen inconsistency 2018-09-28T16:42:17Z <p>Zetapology: /* Reinhardt cardinal */ Has not been refuted in ZFC, but ZFC_2</p> <hr /> <div>{{DISPLAYTITLE:The Kunen inconsistency}}<br /> The Kunen inconsistency, the theorem showing that there can be no nontrivial [[elementary embedding]] from the universe to itself, remains a focal point of large cardinal set theory, marking a hard upper bound at the summit of the main ascent of the large cardinal hierarchy, the first outright refutation of a large cardinal axiom. On this main ascent, large cardinal axioms assert the existence of elementary embeddings $j:V\to M$ where $M$ exhibits increasing affinity with $V$ as one climbs the hierarchy. The $\theta$-[[strong]] cardinals, for example, have $V_\theta\subset M$; the $\lambda$-[[supercompact]] cardinals have $M^\lambda\subset M$; and the [[huge]] cardinals have $M^{j(\kappa)}\subset M$. The natural limit of this trend, first suggested by Reinhardt, is a nontrivial elementary embedding $j:V\to V$, the critical point of which is accordingly known as a ''Reinhardt''<br /> cardinal. Shortly after this idea was introduced, however, Kunen famously proved that there are no such embeddings, and hence no Reinhardt cardinals in $\text{ZFC}$. <br /> <br /> Since that time, the inconsistency argument has been generalized by various authors, including Harada<br /> &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;(p. 320-321),<br /> Hamkins, Kirmayer and Perlmutter &lt;cite&gt;HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency&lt;/cite&gt;, Woodin &lt;cite&gt;Kanamori2009:HigherInfinite&lt;/cite&gt;(p. 320-321),<br /> Zapletal &lt;cite&gt;Zapletal1996:ANewProofOfKunenInconsistency&lt;/cite&gt; and Suzuki &lt;cite&gt;Suzuki1998:NojVtoVinVofG, Suzuki1999:NoDefinablejVtoVinZF&lt;/cite&gt;.<br /> <br /> * There is no nontrivial elementary embedding $j:V\to V$ from the set-theoretic universe to itself.<br /> * There is no nontrivial elementary embedding $j:V[G]\to V$ of a set-forcing extension of the universe to the universe, and neither is there $j:V\to V[G]$ in the converse direction.<br /> * More generally, there is no nontrivial elementary embedding between two ground models of the universe.<br /> * More generally still, there is no nontrivial elementary embedding $j:M\to N$ when both $M$ and $N$ are eventually stationary correct.<br /> * There is no nontrivial elementary embedding $j:V\to \text{HOD}$, and neither is there $j:V\to M$ for a variety of other definable classes, including $\text{gHOD}$ and the $\text{HOD}^\eta$, $\text{gHOD}^\eta$.<br /> * If $j:V\to M$ is elementary, then $V=\text{HOD}(M)$.<br /> * There is no nontrivial elementary embedding $j:\text{HOD}\to V$.<br /> * More generally, for any definable class $M$, there is no nontrivial elementary embedding $j:M\to V$.<br /> * There is no nontrivial elementary embedding $j:\text{HOD}\to\text{HOD}$ that is definable in $V$ from parameters.<br /> <br /> It is not currently known whether the Kunen inconsistency may be undertaken in ZF. Nor is it known whether one may rule out nontrivial embeddings $j:\text{HOD}\to\text{HOD}$ even in $\text{ZFC}$.<br /> <br /> == Metamathematical issues ==<br /> <br /> Kunen formalized his theorem in Kelly-Morse set theory, but it is also possble to prove it in the weaker system of G&amp;ouml;del-Bernays set theory. In each case, the embedding $j$ is a $\text{GBC}$ class, and elementary of $j$ is asserted as a $\Sigma_1$-elementary embedding, which implies $\Sigma_n$-elementarity when the two models have the ordinals.<br /> <br /> == Reinhardt cardinal ==<br /> <br /> Although the existence of Reinhardt cardinals has now been refuted in $\text{ZFC}_2$ and $\text{GBC}$, the term is used in the $\text{ZF}_2$ context to refer to the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.<br /> <br /> == Super Reinhardt cardinal ==<br /> <br /> A ''super Reinhardt'' cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.<br /> <br /> <br /> {{References}}</div> Zetapology