http://cantorsattic.info/api.php?action=feedcontributions&user=Jdh&feedformat=atomCantor's Attic - User contributions [en]2019-11-18T08:40:42ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Upper_attic&diff=1187Upper attic2014-10-02T21:42:14Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Berkeley]] cardinal<br />
* [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Kunen_inconsistency&diff=1186Kunen inconsistency2014-10-02T21:40:36Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The Kunen inconsistency}}<br />
<br />
The Kunen inconsistency, the theorem showing that there can be no nontrivial<br />
elementary embedding from the universe to itself, remains a<br />
focal point of large cardinal set theory, marking a hard<br />
upper bound at the summit of the main ascent of the large<br />
cardinal hierarchy, the first outright refutation of a<br />
large cardinal axiom. On this main ascent, large cardinal<br />
axioms assert the existence of elementary embeddings<br />
$j:V\to M$ where $M$ exhibits increasing affinity with $V$<br />
as one climbs the hierarchy. The $\theta$-[[strong]] cardinals,<br />
for example, have $V_\theta\subset M$; the $\lambda$-[[supercompact]] cardinals have $M^\lambda\subset M$; and<br />
the [[huge]] cardinals have $M^{j(\kappa)}\subset M$. The natural<br />
limit of this trend, first suggested by Reinhardt, is a<br />
nontrivial elementary embedding $j:V\to V$, the critical<br />
point of which is accordingly known as a ''Reinhardt''<br />
cardinal. Shortly after this idea was introduced, however,<br />
Kunen famously proved that there are no such embeddings,<br />
and hence no Reinhardt cardinals in ZFC. <br />
<br />
Since that time, the inconsistency argument has been generalized by various authors, including Harada<br />
<cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),<br />
Hamkins, Kirmayer and Perlmutter <cite>HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency</cite>, Woodin <cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),<br />
Zapletal <cite>Zapletal1996:ANewProofOfKunenInconsistency</cite> and Suzuki <cite>Suzuki1998:NojVtoVinVofG, Suzuki1999:NoDefinablejVtoVinZF</cite>.<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 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 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&ouml;del-Bernays set theory. In each case, the embedding $j$ is a 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 ZFC and GBC, the term is used in the ZF 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>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1185Upper attic2014-10-02T21:38:44Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Berkeley]] cardinal<br />
* [[Reinhardt | super Reinhardt]] cardinal, [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Berkeley&diff=1184Berkeley2014-10-02T21:37:10Z<p>Jdh: </p>
<hr />
<div>A cardinal $\kappa$ is a ''Berkeley'' cardinal, if for any transitive set $M$ with $\kappa\in M$, there is an elementary embedding $j:M\to M$ having critical point less than $\kappa$. These cardinals are defined in the context of ZF set theory without the axiom of choice.<br />
<br />
The Berkeley cardinals were defined by W. Hugh Woodin in about 1992 at his set-theory seminar in Berkeley, with J. D. Hamkins, A. Lewis, D. Seabold, G. Hjorth and perhaps R. Solovay in the audience, among others, issued as a challenge to refute a seemingly over-strong large cardinal axiom. Nevertheless, the existence of these cardinals remains unrefuted in ZF. <br />
<br />
Various strengthenings of the axiom are obtained by imposing conditions on the cofinality of $\kappa$.</div>Jdhhttp://cantorsattic.info/index.php?title=Berkeley&diff=1183Berkeley2014-10-02T21:36:34Z<p>Jdh: Created page with "A cardinal $\kappa$ is a *Berkeley* cardinal, if for any transitive set $M$ with $\kappa\in M$, there is an elementary embedding $j:M\to M$ having critical point less than $\k..."</p>
<hr />
<div>A cardinal $\kappa$ is a *Berkeley* cardinal, if for any transitive set $M$ with $\kappa\in M$, there is an elementary embedding $j:M\to M$ having critical point less than $\kappa$. These cardinals are defined in the context of ZF set theory without the axiom of choice.<br />
<br />
The Berkeley cardinals were defined by W. Hugh Woodin in about 1992 at his set-theory seminar in Berkeley, with J. D. Hamkins, A. Lewis, D. Seabold, G. Hjorth and perhaps R. Solovay in the audience, among others, issued as a challenge to refute a seemingly over-strong large cardinal axiom. Nevertheless, the existence of these cardinals remains unrefuted in ZF. <br />
<br />
Various strengthenings of the axiom are obtained by imposing conditions on the cofinality of $\kappa$.</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1182Upper attic2014-10-02T21:29:09Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Berkeley]] cardinal<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Talk:PFA&diff=1171Talk:PFA2014-03-28T02:00:14Z<p>Jdh: </p>
<hr />
<div>I am not convinced the definition is what we want. Wouldn't every cardinal above the least PFA cardinal also be a PFA cardin<br />
<br />
Yes, that seems to be right. I suppose we want that the forcing is nontrivial unbounded in kappa. Or we could say in the iteration that if PFA holds at that stage, then we add a Cohen real. (by the way, type four tildes to sign your talk posts.) [[User:Jdh|JDH]] ([[User talk:Jdh|talk]]) 19:00, 27 March 2014 (PDT)</div>Jdhhttp://cantorsattic.info/index.php?title=PFA&diff=1167PFA2014-03-25T02:21:37Z<p>Jdh: </p>
<hr />
<div>A cardinal $\kappa$ is a ''PFA cardinal'' if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them. <br />
<br />
Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true.</div>Jdhhttp://cantorsattic.info/index.php?title=PFA&diff=1166PFA2014-03-25T02:19:28Z<p>Jdh: Created page with "A cardinal $\kappa$ is a *PFA cardinal* if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each ..."</p>
<hr />
<div>A cardinal $\kappa$ is a *PFA cardinal* if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them. <br />
<br />
Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true.</div>Jdhhttp://cantorsattic.info/index.php?title=Worldly&diff=1165Worldly2014-03-25T02:14:25Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of ZFC. 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 worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1164Upper attic2014-03-25T02:13:36Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1163Upper attic2014-03-25T02:07:48Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[iterable#1-iterable | $1$-iterable]] cardinal, and the [[iterable#alpha-iterable | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1162Upper attic2014-03-25T02:05:42Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1161Upper attic2014-03-25T02:03:50Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[PFA]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Worldly&diff=1160Worldly2014-03-25T02:02:05Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of ZFC. 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 worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center.</div>Jdhhttp://cantorsattic.info/index.php?title=Worldly&diff=1159Worldly2014-03-25T02:01:05Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of ZFC. 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 />
<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 worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center.</div>Jdhhttp://cantorsattic.info/index.php?title=User:Jdh&diff=1158User:Jdh2014-03-25T01:59:53Z<p>Jdh: </p>
<hr />
<div>[http://jdh.hamkins.org Joel David Hamkins] is a professor of mathematics at the City University of New York (College of Staten Island and the CUNY Graduate Center), conducting research in mathematical logic, particularly in set theory.</div>Jdhhttp://cantorsattic.info/index.php?title=Parlour&diff=1139Parlour2013-07-30T20:15:32Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The parlour}}<br />
[[File:PTLomaLighthouseByWagbogwest.jpg | thumb | PT Loma Lighthouse photo by wagbogwest]]<br />
<br />
Welcome to the parlour, where finite numbers dream. <br />
<br />
* up to [[omega]] and into the [[lower attic | attic]]<br />
* over to [http://modelsofpa.info Peano's Parlour], the site for open problems in nonstandard models of Peano Arithmetic and related theories<br />
* The smallest non-describable number<br />
* The smallest uninteresting number<br />
* [[Busy beaver function]] values<br />
* [[Fast-growing hierarchy]]<br />
* [[Ackerman function]] values<br />
* [[Graham | Graham's number]]<br />
* [[Ramsey numbers]]<br />
* [[googol#Googol_plex | googol plex]], [[googol#Googol_bang | googol bang]], [[googol#Googol_stack | googol stack]] and the [[googol#The_googol_plex_bang_stack_hierarchy | googol plex bang stack hierarchy]]<br />
* [[googol]]<br />
* 42<br />
* 1<br />
* [[Zero | 0]]</div>Jdhhttp://cantorsattic.info/index.php?title=Parlour&diff=1138Parlour2013-07-30T20:14:13Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The parlour}}<br />
[[File:PTLomaLighthouseByWagbogwest.jpg | thumb | PT Loma Lighthouse photo by wagbogwest]]<br />
<br />
Welcome to the parlour, where finite numbers dream. <br />
<br />
* up to [[omega]] and into the [[lower attic | attic]]<br />
* over to [[Peano's Parlour]], the site for open problems in nonstandard models of Peano Arithmetic and related theories<br />
* The smallest non-describable number<br />
* The smallest uninteresting number<br />
* [[Busy beaver function]] values<br />
* [[Fast-growing hierarchy]]<br />
* [[Ackerman function]] values<br />
* [[Graham | Graham's number]]<br />
* [[Ramsey numbers]]<br />
* [[googol#Googol_plex | googol plex]], [[googol#Googol_bang | googol bang]], [[googol#Googol_stack | googol stack]] and the [[googol#The_googol_plex_bang_stack_hierarchy | googol plex bang stack hierarchy]]<br />
* [[googol]]<br />
* 42<br />
* 1<br />
* [[Zero | 0]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1137Upper attic2013-07-29T03:00:55Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<br />
* [[Kelly-Morse]] 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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Omega_one_chess&diff=1136Omega one chess2013-07-27T18:21:38Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The omega one of chess}}<br />
<br />
Infinite chess is chess played on an infinite edgeless chessboard. Since checkmates, when they occur, do so after finitely many moves, this is an open game and therefore subject to the theory of transfinite game values for open games. <br />
<br />
Specifically, the ''game value'' (for white) of a position in infinite chess is defined by recursion. The positions with value $0$ are precisely those in which white has already won. If a position $p$ has white to move, then the value of $p$ is $\alpha+1$ if and only if $\alpha$ is minimal such that white may legally move from $p$ to a position with value $\alpha$. If a position $p$ has black to play, where black has a legal move from $p$, and every move by black from $p$ has a value, then the value of $p$ is the supremum of these values.<br />
<br />
The term ''omega one of chess'' refers either to the ordinal $\omega_1^{\mathfrak{Ch}}$ or to $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, depending respectively on whether one is considering only finite positions or also positions with infinitely many pieces.<cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite><br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}}$ is the supremum of the game values of the winning finite positions for white in infinite chess.<br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ is the supremum of the game values of all the winning positions for white in infinite chess, including positions with infinitely many pieces.<br />
<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}$ and $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}$ are the analogous ordinals for infinite three-dimensional chess, as described in .<br />
<br />
There is an entire natural hierarchy of intermediate concepts between $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, corresponding to the various descriptive-set-theoretic complexities of the positions. For example, we may denote by $\omega_1^{\mathfrak{Ch},c}$ the 'computable' omega one of chess, which is the supremum of the game values of the computable positions of infinite chess. Similarly, one may define $\omega_1^{\mathfrak{Ch},\text{hyp}}$ to be the supremum of the values of the hyperarithmetic positions and $\omega_1^{\mathfrak{Ch},\Delta^1_2}$ to be the supremum of the $\Delta^1_2$ definable positions, and so on.<br />
<br />
Since there are only countably many finite positions, whose game values form an initial segment of the ordinals, it follows that $\omega_1^{\mathfrak{Ch}}$ is a countable ordinal. The next theorem shows more, that $\omega_1^{\mathfrak{Ch}}$ is bounded by the [[Church-Kleene]] ordinal $\omega_1^{ck}$, the supremum of the computable ordinals. Thus, the game value of any finite position in infinite chess with a game value is a computable ordinal.<br />
<br />
* $\omega_1^{\mathfrak{Ch}}\leq\omega_1^{\mathfrak{Ch},c}\leq\omega_1^{\mathfrak{Ch},\text{hyp}}\leq\omega_1^{ck}$. Thus, the game value of any winning finite position in infinite chess, as well as any winning computable position or winning hyperarithmetic position, is a computable ordinal. Furthermore, if a designated player has a winning strategy from a position $p$, then there is such a strategy with complexity hyperarithmetic in $p$.<br />
* $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\leq\omega_1$. The value of a winning position $p$ is a $p$-computable ordinal, and hence less than $\omega_1^{ck,p}$.<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}\leq\omega_1^{ck}$. <br />
<br />
Evans and Hamkins <cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite> have proved that the omega one of infinite 3D chess is true $\omega_1$, as large as it can be: $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}=\omega_1$. Meanwhile, there remains a large gap in the best-known bounds for $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, with Evans and Hamkins finding (computable) infinite positions having value $\omega^3$ and somewhat beyond. <br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Lower_attic&diff=1135Lower attic2013-07-27T14:14:30Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The lower attic}}<br />
[[File:SagradaSpiralByDavidNikonvscanon.jpg | thumb | Sagrada Spiral photo by David Nikonvscanon]]<br />
<br />
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.<br />
<br />
* [[aleph one | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]<br />
* [[stable]] ordinals<br />
* The ordinals of [[infinite time Turing machines]], including <br />
** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals<br />
** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals <br />
** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals, <br />
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]<br />
* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals<br />
* the [[Bachmann-Howard]] ordinal<br />
* the [[large Veblen]] ordinal<br />
* the [[small Veblen]] ordinal<br />
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]<br />
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]<br />
* the [[omega one chess | omega one of chess]]<br />
** [[omega one chess|$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$]] = the supremum of the game values for white of all positions in infinite chess<br />
** [[omega one chess| $\omega_1^{\mathfrak{Ch},c}$]] = the supremum of the game values for white of the computable positions in infinite chess <br />
** [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]] = the supremum of the game values for white of the finite positions in infinite chess<br />
* [[indecomposable]] ordinal<br />
* the [[small countable ordinals]], such as [[small countable ordinals | $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$]] up to [[epsilon naught | $\epsilon_0$]] <br />
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]<br />
* [[omega | $\omega$]], the smallest infinity<br />
* down to the [[parlour]], where large finite numbers dream</div>Jdhhttp://cantorsattic.info/index.php?title=Omega_one_chess&diff=1134Omega one chess2013-07-27T13:59:53Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The omega one of chess}}<br />
<br />
Infinite chess is chess played on an infinite edgeless chessboard. Since checkmates, when they occur, do so after finitely many moves, this is an open game and therefore subject to the theory of transfinite game values for open games. <br />
<br />
Specifically, the 'game value' (for white) of a position in infinite chess is defined by recursion. The positions with value $0$ are precisely those in which white has already won. If a position $p$ has white to move, then the value of $p$ is $\alpha+1$ if and only if $\alpha$ is minimal such that white may legally move from $p$ to a position with value $\alpha$. If a position $p$ has black to play, where black has a legal move from $p$, and every move by black from $p$ has a value, then the value of $p$ is the supremum of these values.<br />
<br />
The term ``omega one of chess'' refers either to the ordinal $\omega_1^{\mathfrak{Ch}}$ or to $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, depending respectively on whether one is considering only finite positions or also positions with infinitely many pieces.<cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite><br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}}$ is the supremum of the game values of the winning finite positions for white in infinite chess.<br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ is the supremum of the game values of all the winning positions for white in infinite chess, including positions with infinitely many pieces.<br />
<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}$ and $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}$ are the analogous ordinals for infinite three-dimensional chess, as described in .<br />
<br />
There is an entire natural hierarchy of intermediate concepts between $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, corresponding to the various descriptive-set-theoretic complexities of the positions. For example, we may denote by $\omega_1^{\mathfrak{Ch},c}$ the 'computable' omega one of chess, which is the supremum of the game values of the computable positions of infinite chess. Similarly, one may define $\omega_1^{\mathfrak{Ch},\text{hyp}}$ to be the supremum of the values of the hyperarithmetic positions and $\omega_1^{\mathfrak{Ch},\Delta^1_2}$ to be the supremum of the $\Delta^1_2$ definable positions, and so on.<br />
<br />
Since there are only countably many finite positions, whose game values form an initial segment of the ordinals, it follows that $\omega_1^{\mathfrak{Ch}}$ is a countable ordinal. The next theorem shows more, that $\omega_1^{\mathfrak{Ch}}$ is bounded by the [[Church-Kleene]] ordinal $\omega_1^{ck}$, the supremum of the computable ordinals. Thus, the game value of any finite position in infinite chess with a game value is a computable ordinal.<br />
<br />
* $\omega_1^{\mathfrak{Ch}}\leq\omega_1^{\mathfrak{Ch},c}\leq\omega_1^{\mathfrak{Ch},\text{hyp}}\leq\omega_1^{ck}$. Thus, the game value of any winning finite position in infinite chess, as well as any winning computable position or winning hyperarithmetic position, is a computable ordinal. Furthermore, if a designated player has a winning strategy from a position $p$, then there is such a strategy with complexity hyperarithmetic in $p$.<br />
* $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\leq\omega_1$. The value of a winning position $p$ is a $p$-computable ordinal, and hence less than $\omega_1^{ck,p}$.<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}\leq\omega_1^{ck}$. <br />
<br />
Evans and Hamkins <cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite> have proved that the omega one of infinite 3D chess is true $\omega_1$, as large as it can be: $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}=\omega_1$. Meanwhile, there remains a large gap in the best-known bounds for $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, with Evans and Hamkins finding (computable) infinite positions having value $\omega^3$. <br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Lower_attic&diff=1133Lower attic2013-07-27T13:57:51Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The lower attic}}<br />
[[File:SagradaSpiralByDavidNikonvscanon.jpg | thumb | Sagrada Spiral photo by David Nikonvscanon]]<br />
<br />
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.<br />
<br />
* [[aleph one | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]<br />
* [[stable]] ordinals<br />
* The ordinals of [[infinite time Turing machines]], including <br />
** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals<br />
** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals <br />
** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals, <br />
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]<br />
* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals<br />
* the [[Bachmann-Howard]] ordinal<br />
* the [[large Veblen]] ordinal<br />
* the [[small Veblen]] ordinal<br />
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]<br />
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]<br />
* the [[omega one chess | omega one of chess]], [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]], [[omega one chess|$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$]]<br />
* [[indecomposable]] ordinal<br />
* the [[small countable ordinals]], such as [[small countable ordinals | $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$]] up to [[epsilon naught | $\epsilon_0$]] <br />
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]<br />
* [[omega | $\omega$]], the smallest infinity<br />
* down to the [[parlour]], where large finite numbers dream</div>Jdhhttp://cantorsattic.info/index.php?title=Omega_one_chess&diff=1132Omega one chess2013-07-27T13:38:03Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The omega one of infinite chess}}<br />
<br />
Infinite chess is chess played on an infinite edgeless chessboard. Since checkmates, when they occur, do so after finitely many moves, this is an open game and therefore subject to the theory of transfinite game values for open games. <br />
<br />
Specifically, the 'game value' (for white) of a position in infinite chess is defined by recursion. The positions with value $0$ are precisely those in which white has already won. If a position $p$ has white to move, then the value of $p$ is $\alpha+1$ if and only if $\alpha$ is minimal such that white may legally move from $p$ to a position with value $\alpha$. If a position $p$ has black to play, where black has a legal move from $p$, and every move by black from $p$ has a value, then the value of $p$ is the supremum of these values.<br />
<br />
The term ``omega one of chess'' refers either to the ordinal $\omega_1^{\mathfrak{Ch}}$ or to $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, depending respectively on whether one is considering only finite positions or also positions with infinitely many pieces.<cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite><br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}}$ is the supremum of the game values of the winning finite positions for white in infinite chess.<br />
<br />
* The ordinal $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ is the supremum of the game values of all the winning positions for white in infinite chess, including positions with infinitely many pieces.<br />
<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}$ and $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}$ are the analogous ordinals for infinite three-dimensional chess, as described in .<br />
<br />
There is an entire natural hierarchy of intermediate concepts between $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, corresponding to the various descriptive-set-theoretic complexities of the positions. For example, we may denote by $\omega_1^{\mathfrak{Ch},c}$ the 'computable' omega one of chess, which is the supremum of the game values of the computable positions of infinite chess. Similarly, one may define $\omega_1^{\mathfrak{Ch},\text{hyp}}$ to be the supremum of the values of the hyperarithmetic positions and $\omega_1^{\mathfrak{Ch},\Delta^1_2}$ to be the supremum of the $\Delta^1_2$ definable positions, and so on.<br />
<br />
Since there are only countably many finite positions, whose game values form an initial segment of the ordinals, it follows that $\omega_1^{\mathfrak{Ch}}$ is a countable ordinal. The next theorem shows more, that $\omega_1^{\mathfrak{Ch}}$ is bounded by the [[Church-Kleene]] ordinal $\omega_1^{ck}$, the supremum of the computable ordinals. Thus, the game value of any finite position in infinite chess with a game value is a computable ordinal.<br />
<br />
* $\omega_1^{\mathfrak{Ch}}\leq\omega_1^{\mathfrak{Ch},c}\leq\omega_1^{\mathfrak{Ch},\text{hyp}}\leq\omega_1^{ck}$. Thus, the game value of any winning finite position in infinite chess, as well as any winning computable position or winning hyperarithmetic position, is a computable ordinal. Furthermore, if a designated player has a winning strategy from a position $p$, then there is such a strategy with complexity hyperarithmetic in $p$.<br />
* $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\leq\omega_1$. The value of a winning position $p$ is a $p$-computable ordinal, and hence less than $\omega_1^{ck,p}$.<br />
* Similarly, $\omega_1^{\mathfrak{Ch}_3}\leq\omega_1^{ck}$. <br />
<br />
Evans and Hamkins <cite>EvansHamkins:TransfiniteGameValuesInInfiniteChess</cite> have proved that the omega one of infinite 3D chess is true $\omega_1$, as large as it can be: $\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}=\omega_1$. Meanwhile, there remains a large gap in the best-known bounds for $\omega_1^{\mathfrak{Ch}}$ and $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, with Evans and Hamkins finding (computable) infinite positions having value $\omega^3$. <br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1131Library2013-07-27T12:39:20Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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 = "Definable orthogonality classes in accessible categories are small",<br />
NOTE = "submitted for publication",<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&#337;s on his 60th birthday), Vol. I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency, <br />
author = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman},<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic},<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<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 />
#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 />
}<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1130Library2013-07-24T01:48:17Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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 = "Definable orthogonality classes in accessible categories are small",<br />
NOTE = "submitted for publication",<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&#337;s on his 60th birthday), Vol. I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency, <br />
author = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman},<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic},<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<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 />
#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 />
}<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1129Library2013-07-24T01:45:46Z<p>Jdh: /* Library holdings */</p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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<br />
Mathias, A. R. D. and Rosicky, Jirí},<br />
TITLE = "Definable orthogonality classes in accessible categories are<br />
small",<br />
NOTE = "submitted for publication",<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;<br />
dedicated to P. Erd&#337;s on his 60th birthday), Vol.<br />
I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency, <br />
author = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman},<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic},<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<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 />
#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 />
}<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Woodin&diff=1128Woodin2013-07-24T01:37:54Z<p>Jdh: /* Role in $\Omega$-Logic and the Resurrection Theorem */</p>
<hr />
<div>{{DISPLAYTITLE:Woodin Cardinal}}<br />
<br />
Woodin cardinals are a generalization of the notion of [[strong]] cardinals and have been used to calibrate the exact proof-theoretic strength of the [[Axiom of Determinacy]]. Woodin cardinals are weaker than [[superstrong]] cardinals in consistency strength and fail to be [[weakly compact]] in general, since they are not $\Pi_1^1$ [[indescribable]]. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed. <br />
<br />
==Shelah cardinals==<br />
<br />
Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc.). In slightly more detail, Woodin had established that the Axiom of Determinacy (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a non-trivial elementary embedding [[L of V lambda+1 |$j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] with critical point $<\lambda$. This axiom is known to be very strong and its use was first weakened to that of the existence of a [[supercompact]] cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the regularity properties of sets of reals mentioned above. <br />
<br />
A cardinal $\kappa$ is Shelah if, for every function $f:\kappa\to\kappa$ there is a non-trivial elementary embedding $j:V\prec M$ with $M$ a transitive class, $\kappa$ the critical point of $j$ and $M$ contains the initial segment $V_{j(f)(\kappa)}$.<br />
<br />
Shelah cardinals are above [[strong]] but below [[superstrong]] in the large cardinal/consistency hierarchy.<br />
<br />
It turns out that Shelah cardinals have many large cardinals below them that suffice to establish the regularity properties, and as a result have mostly faded from view in the large cardinal research literature. <br />
<br />
==Woodin cardinals==<br />
<br />
Woodin cardinals are a refinement of Shelah cardinals. The primary difference is the requirement of a closure condition on the functions $f:\kappa\rightarrow\kappa$ and associated embeddings. Woodin cardinals are not themselves the critical points of any of their associated embeddings and hence need not be measurable. They are, however, Mahlo cardinals (and hence also inaccessible) since the set of measurable cardinals below a Woodin cardinal must be stationary. <br />
<br />
<br />
<br />
=== Elementary Embedding Characterization ===<br />
<br />
A cardinal $\kappa$ is "Woodin" if, for every function $f:\kappa\to\kappa$ there is some $\gamma<\kappa$ such that $f$ is closed under $\gamma$ and there is an associated non-trivial elementary embedding $j:V\prec M$ with critical point $\gamma$ where $M$ contains the initial segment $V_{(j(f))(\gamma)}$. <br />
<br />
If $\kappa$ is Woodin then for any subset $A\subseteq V_\kappa$ some $\alpha <\kappa$ is $\gamma$-strong for every $\gamma <\kappa$. Intuitively this means that there are elementary embeddings $j_\gamma$ which preserve $A$ i.e., $A\cap V_{\alpha+\gamma}=j_\gamma(A)\cap V_{\alpha+\gamma}$, have critical point $\alpha$, and whose target transitive class contains the initial segment $V_{\alpha+\gamma}$. <br />
<br />
There is a hierarchy of Woodin-type cardinals<br />
<br />
==Analogue of Vopěnka's Principle==<br />
<br />
This material will be added later.<br />
<br />
See also the [[Woodin for supercompactness]] cardinals, which are identical to the [[Vopenka | Vopěnka]] cardinals.<br />
<br />
==Stationary Tower Forcing==<br />
<br />
==Role in $\Omega$-Logic and the Resurrection Theorem==<br />
<br />
<nocite>Jech2003:SetTheory</nocite><br />
<br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Shelah&diff=1127Shelah2013-07-24T01:36:53Z<p>Jdh: Redirected page to Woodin#Shelah cardinals</p>
<hr />
<div>#REDIRECT [[Woodin#Shelah cardinals]]</div>Jdhhttp://cantorsattic.info/index.php?title=Woodin&diff=1126Woodin2013-07-24T01:34:20Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:Woodin Cardinal}}<br />
<br />
Woodin cardinals are a generalization of the notion of [[strong]] cardinals and have been used to calibrate the exact proof-theoretic strength of the [[Axiom of Determinacy]]. Woodin cardinals are weaker than [[superstrong]] cardinals in consistency strength and fail to be [[weakly compact]] in general, since they are not $\Pi_1^1$ [[indescribable]]. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed. <br />
<br />
==Shelah cardinals==<br />
<br />
Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc.). In slightly more detail, Woodin had established that the Axiom of Determinacy (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a non-trivial elementary embedding [[L of V lambda+1 |$j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] with critical point $<\lambda$. This axiom is known to be very strong and its use was first weakened to that of the existence of a [[supercompact]] cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the regularity properties of sets of reals mentioned above. <br />
<br />
A cardinal $\kappa$ is Shelah if, for every function $f:\kappa\to\kappa$ there is a non-trivial elementary embedding $j:V\prec M$ with $M$ a transitive class, $\kappa$ the critical point of $j$ and $M$ contains the initial segment $V_{j(f)(\kappa)}$.<br />
<br />
Shelah cardinals are above [[strong]] but below [[superstrong]] in the large cardinal/consistency hierarchy.<br />
<br />
It turns out that Shelah cardinals have many large cardinals below them that suffice to establish the regularity properties, and as a result have mostly faded from view in the large cardinal research literature. <br />
<br />
==Woodin cardinals==<br />
<br />
Woodin cardinals are a refinement of Shelah cardinals. The primary difference is the requirement of a closure condition on the functions $f:\kappa\rightarrow\kappa$ and associated embeddings. Woodin cardinals are not themselves the critical points of any of their associated embeddings and hence need not be measurable. They are, however, Mahlo cardinals (and hence also inaccessible) since the set of measurable cardinals below a Woodin cardinal must be stationary. <br />
<br />
<br />
<br />
=== Elementary Embedding Characterization ===<br />
<br />
A cardinal $\kappa$ is "Woodin" if, for every function $f:\kappa\to\kappa$ there is some $\gamma<\kappa$ such that $f$ is closed under $\gamma$ and there is an associated non-trivial elementary embedding $j:V\prec M$ with critical point $\gamma$ where $M$ contains the initial segment $V_{(j(f))(\gamma)}$. <br />
<br />
If $\kappa$ is Woodin then for any subset $A\subseteq V_\kappa$ some $\alpha <\kappa$ is $\gamma$-strong for every $\gamma <\kappa$. Intuitively this means that there are elementary embeddings $j_\gamma$ which preserve $A$ i.e., $A\cap V_{\alpha+\gamma}=j_\gamma(A)\cap V_{\alpha+\gamma}$, have critical point $\alpha$, and whose target transitive class contains the initial segment $V_{\alpha+\gamma}$. <br />
<br />
There is a hierarchy of Woodin-type cardinals<br />
<br />
==Analogue of Vopěnka's Principle==<br />
<br />
This material will be added later.<br />
<br />
See also the [[Woodin for supercompactness]] cardinals, which are identical to the [[Vopenka | Vopěnka]] cardinals.<br />
<br />
==Stationary Tower Forcing==<br />
<br />
==Role in $\Omega$-Logic and the Resurrection Theorem==<br />
<br />
<cite>Jech2003:SetTheory</cite><br />
<br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1125Library2013-07-24T01:29:35Z<p>Jdh: /* Library holdings */</p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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<br />
Mathias, A. R. D. and Rosicky, Jirí},<br />
TITLE = "Definable orthogonality classes in accessible categories are<br />
small",<br />
NOTE = "submitted for publication",<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;<br />
dedicated to P. Erd&#337;s on his 60th birthday), Vol.<br />
I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency, <br />
author = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman},<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic},<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<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 />
#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 />
}<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Superstrong&diff=1124Superstrong2013-07-24T01:22:01Z<p>Jdh: Redirected page to Strong#Superstrong cardinal</p>
<hr />
<div>#REDIRECT [[strong#Superstrong cardinal]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1123Upper attic2013-07-24T01:20:56Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1122Upper attic2013-07-24T01:13:56Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[strong#Superstrong cardinal| superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1121Upper attic2013-07-24T01:12:49Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[strong#Superstrong cardinal| superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1120Upper attic2013-07-24T01:09:26Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[strong#Superstrong cardinal| superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Vopenka&diff=1119Vopenka2013-07-24T01:08:25Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Vopěnka's principle and Vopěnka cardinals}}<br />
<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 />
<blockquote><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 />
</blockquote><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]], then $V_\kappa$ satisfies Vopěnka's principle. <br />
<br />
==Formalisations==<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 $\kappa$. <br />
<br />
Perlmutter <cite>Perlmutter:Dissertation</cite> proved that a cardinal is a Vopěnka cardinal if and only if it is a [[Woodin for supercompactness]] cardinal. <br />
<br />
As we mentioned above, every almost huge cardinal is a Vopěnka cardinal. <br />
<br />
==Equivalent statements==<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<br />
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 />
<CITE>BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses</CITE><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<br />
over $V_\kappa$, and $\Pi^1_1$ 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>Jdhhttp://cantorsattic.info/index.php?title=Vopenka&diff=1118Vopenka2013-07-24T00:58:50Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Vopěnka's principle and Vopěnka cardinals}}<br />
<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 />
<blockquote><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 />
</blockquote><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]], then $V_\kappa$ satisfies Vopěnka's principle.<br />
<br />
==Formalisations==<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.<br />
One alternative is to view Vopěnka's principle as an axiom in a class theory, such as von Neumann-Gödel-Bernays. Another is to consider a '''Vopěnka cardinal''', that is, a cardinal<br />
$\kappa$ that is inaccessible and such that $V_\kappa$ satisfies Vopěnka's principle when "proper class" is taken to mean "subset of $V_\kappa$ of cardinality $\kappa$.<br />
These three alternatives are, in the order listed, strictly increasing in strength [http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably].<br />
<br />
==Equivalent statements==<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<br />
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 />
<CITE>BagariaCasacubertaMathiasRosicky2012:OrthogonalityClasses</CITE><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<br />
over $V_\kappa$, and $\Pi^1_1$ 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>Jdhhttp://cantorsattic.info/index.php?title=Upper_attic&diff=1117Upper attic2013-07-24T00:54:20Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<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]]<br />
* [[Reinhardt]] cardinal<br />
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]<br />
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$ <br />
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$<br />
* The [[wholeness axiom]]<br />
* [[huge#Super_n-huge | super $n$-huge]] cardinal<br />
* [[huge#Superhuge | superhuge]] cardinal<br />
* [[huge]] cardinal<br />
* [[huge#Almost huge | almost huge]] cardinal<br />
* [[Vopenka#Formalisations | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal<br />
* [[grand reflection]] cardinal<br />
* [[supercompact]] cardinal<br />
* [[strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[strong#Superstrong cardinal| superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* [[Woodin]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy<br />
* [[tall]] cardinal<br />
* [[zero dagger| $0^\dagger$]]<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[measurable]] cardinal<br />
* [[weakly measurable]] cardinal<br />
* [[strongly Ramsey]] cardinal<br />
* [[Ramsey]] cardinal<br />
* [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* [[zero sharp | $0^\sharp$]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[completely ineffable]] cardinal<br />
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy<br />
* [[weakly ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, , [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[unfoldable]] cardinal<br />
* [[Totally indescribable]] cardinal<br />
* [[indescribable]] cardinal<br />
* [[weakly compact]] cardinal<br />
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals <br />
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy <br />
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]<br />
* [[Mahlo]] cardinal<br />
* [[uplifting]] cardinal<br />
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal<br />
* [[reflecting]] cardinal<br />
* [[ORD is Mahlo]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<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, also known as strongly inaccessible<br />
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal<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]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] <br />
<br />
* down to [[the middle attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Uplifting&diff=1116Uplifting2013-07-23T18:26:33Z<p>Jdh: /* Connection with the resurrection axioms */</p>
<hr />
<div>{{DISPLAYTITLE: Uplifting cardinals}}<br />
<br />
Uplifting cardinals were introduced by Hamkins and Johnstone in <cite>HamkinsJohnstone:ResurrectionAxioms</cite>, from which some of this text is adapted.<br />
<br />
An inaccessible cardinal $\kappa$ is ''uplifting'' if and only if for every ordinal $\theta$ it is ''$\theta$-uplifting'', meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension. <br />
<br />
An inaccessible cardinal is ''pseudo uplifting'' if and only if for every ordinal $\theta$ it is ''pseudo $\theta$-uplifting'', meaning that there is a cardinal $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension, without insisting that $\gamma$ is inaccessible.<br />
<br />
It is an elementary exercise to see that if $V_\kappa\prec V_\gamma$ is a proper elementary extension, then $\kappa$ and hence also $\gamma$ are [[Beth fixed point | $\beth$-fixed points]], and so $V_\kappa=H_\kappa$ and $V_\gamma=H_\gamma$. It follows that a cardinal $\kappa$ is uplifting if and only if it is regular and there are arbitrarily large regular cardinals $\gamma$ such that $H_\kappa\prec H_\gamma$. It is also easy to see that every uplifting cardinal $\kappa$ is uplifting in $L$, with the same targets. Namely, if $V_\kappa\prec V_\gamma$, then we may simply restrict to the constructible sets to obtain $V_\kappa^L=L^{V_\kappa}\prec L^{V_\gamma}=V_\gamma^L$. An analogous result holds for pseudo-uplifting cardinals.<br />
<br />
== Consistency strength of uplifting cardinals == <br />
<br />
The consistency strength of uplifting and pseudo-uplifting cardinals are bounded between the existence of a [[Mahlo]] cardinal and the hypothesis [[Ord is Mahlo]]. <br />
<br />
'''Theorem.''' <br />
<br />
1. If $\delta$ is a [[Mahlo]] cardinal, then $V_\delta$ has a proper class of uplifting cardinals.<br />
<br />
2. Every uplifting cardinal is pseudo uplifting and a limit of pseudo uplifting cardinals.<br />
<br />
3. If there is a pseudo uplifting cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a [[reflecting]] cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. <br />
<br />
Proof. For (1), suppose that $\delta$ is a Mahlo cardinal. By the Lowenheim-Skolem theorem, there is a club set $C\subset\delta$ of cardinals $\beta$ with $V_\beta\prec V_\delta$. Since $\delta$ is Mahlo, the club $C$ contains unboundedly many inaccessible cardinals. If $\kappa<\gamma$ are both in $C$, then $V_\kappa\prec V_\gamma$, as desired. Similarly, for (2), if $\kappa$ is uplifting, then $\kappa$ is pseudo uplifting and if $V_\kappa\prec V_\gamma$ with $\gamma$ inaccessible, then there are unboundedly many ordinals $\beta<\gamma$ with $V_\beta\prec V_\gamma$ and hence $V_\kappa\prec V_\beta$. So $\kappa$ is pseudo uplifting in $V_\gamma$. From this, it follows that there must be unboundedly many pseudo uplifting cardinals below $\kappa$. For (3), if $\kappa$ is inaccessible and $V_\kappa\prec V_\gamma$, then $V_\gamma$ is a transitive set model of ZFC in which $\kappa$ is reflecting, and it is thus also a model of [[Ord is Mahlo]]. QED<br />
<br />
== Uplifting cardinals and $\Sigma_3$-reflection == <br />
<br />
* Every uplifting cardinal is a limit of $\Sigma_3$-reflecting cardinals, and is itself $\Sigma_3$-reflecting.<br />
* If $\kappa$ is the least uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
The analogous observation for pseudo uplifting cardinals holds as well, namely, every pseudo uplifting cardinal is $\Sigma_3$-reflecting and a limit of $\Sigma_3$-reflecting cardinals; and if $\kappa$ is the least pseudo uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
== Uplifting Laver functions ==<br />
<br />
Every uplifting cardinal admits an ordinal-anticipating Laver function, and indeed, a HOD-anticipating Laver function, a function $\ell:\kappa\to V_\kappa$, definable in $V_\kappa$, such that for any set $x\in\text{HOD}$ and $\theta$, there is an inaccessible cardinal $\gamma$ above $\theta$ such that $V_\kappa\prec V_\gamma$, for which $\ell^*(\kappa)=x$, where $\ell^*$ is the corresponding function defined in $V_\gamma$. <br />
<br />
== Connection with the resurrection axioms ==<br />
<br />
Many instances of the (weak) resurrection axiom imply that ${\frak c}^V$ is an uplifting cardinal in $L$:<br />
* RA(all) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* RA(ccc) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* wRA(countably closed)+$\neg$CH implies that ${\frak c}^V$ is uplifting in $L$.<br />
* Under $\neg$CH, the weak resurrection axioms for the classes of axiom-A forcing, proper forcing, semi-proper forcing, and posets that preserve stationary subsets of $\omega_1$, respectively, each imply that ${\frak c}^V$ is uplifting in $L$.<br />
<br />
Conversely, if $\kappa$ is uplifting, then various resurrection axioms hold in a corresponding lottery-iteration forcing extension. <br />
<br />
'''Theorem.''' (Hamkins and Johnstone) The following theories are equiconsistent over ZFC:<br />
* There is an uplifting cardinal.<br />
* RA(all)<br />
* RA(ccc)<br />
* RA(semiproper)+$\neg$CH<br />
* RA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, RA($\alpha$-proper)+$\neg$CH<br />
* RA(axiom-A)+$\neg$CH<br />
* wRA(semiproper)+$\neg$CH<br />
* wRA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, wRA($\alpha$-proper})+$\neg$CH<br />
* wRA(axiom-A)+$\neg$CH<br />
* wRA(countably closed)+$\neg$CH<br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Uplifting&diff=1115Uplifting2013-07-23T18:26:07Z<p>Jdh: /* Connection with the resurrection axioms */</p>
<hr />
<div>{{DISPLAYTITLE: Uplifting cardinals}}<br />
<br />
Uplifting cardinals were introduced by Hamkins and Johnstone in <cite>HamkinsJohnstone:ResurrectionAxioms</cite>, from which some of this text is adapted.<br />
<br />
An inaccessible cardinal $\kappa$ is ''uplifting'' if and only if for every ordinal $\theta$ it is ''$\theta$-uplifting'', meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension. <br />
<br />
An inaccessible cardinal is ''pseudo uplifting'' if and only if for every ordinal $\theta$ it is ''pseudo $\theta$-uplifting'', meaning that there is a cardinal $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension, without insisting that $\gamma$ is inaccessible.<br />
<br />
It is an elementary exercise to see that if $V_\kappa\prec V_\gamma$ is a proper elementary extension, then $\kappa$ and hence also $\gamma$ are [[Beth fixed point | $\beth$-fixed points]], and so $V_\kappa=H_\kappa$ and $V_\gamma=H_\gamma$. It follows that a cardinal $\kappa$ is uplifting if and only if it is regular and there are arbitrarily large regular cardinals $\gamma$ such that $H_\kappa\prec H_\gamma$. It is also easy to see that every uplifting cardinal $\kappa$ is uplifting in $L$, with the same targets. Namely, if $V_\kappa\prec V_\gamma$, then we may simply restrict to the constructible sets to obtain $V_\kappa^L=L^{V_\kappa}\prec L^{V_\gamma}=V_\gamma^L$. An analogous result holds for pseudo-uplifting cardinals.<br />
<br />
== Consistency strength of uplifting cardinals == <br />
<br />
The consistency strength of uplifting and pseudo-uplifting cardinals are bounded between the existence of a [[Mahlo]] cardinal and the hypothesis [[Ord is Mahlo]]. <br />
<br />
'''Theorem.''' <br />
<br />
1. If $\delta$ is a [[Mahlo]] cardinal, then $V_\delta$ has a proper class of uplifting cardinals.<br />
<br />
2. Every uplifting cardinal is pseudo uplifting and a limit of pseudo uplifting cardinals.<br />
<br />
3. If there is a pseudo uplifting cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a [[reflecting]] cardinal and consequently also a transitive model of ZFC plus [[Ord is Mahlo]]. <br />
<br />
Proof. For (1), suppose that $\delta$ is a Mahlo cardinal. By the Lowenheim-Skolem theorem, there is a club set $C\subset\delta$ of cardinals $\beta$ with $V_\beta\prec V_\delta$. Since $\delta$ is Mahlo, the club $C$ contains unboundedly many inaccessible cardinals. If $\kappa<\gamma$ are both in $C$, then $V_\kappa\prec V_\gamma$, as desired. Similarly, for (2), if $\kappa$ is uplifting, then $\kappa$ is pseudo uplifting and if $V_\kappa\prec V_\gamma$ with $\gamma$ inaccessible, then there are unboundedly many ordinals $\beta<\gamma$ with $V_\beta\prec V_\gamma$ and hence $V_\kappa\prec V_\beta$. So $\kappa$ is pseudo uplifting in $V_\gamma$. From this, it follows that there must be unboundedly many pseudo uplifting cardinals below $\kappa$. For (3), if $\kappa$ is inaccessible and $V_\kappa\prec V_\gamma$, then $V_\gamma$ is a transitive set model of ZFC in which $\kappa$ is reflecting, and it is thus also a model of [[Ord is Mahlo]]. QED<br />
<br />
== Uplifting cardinals and $\Sigma_3$-reflection == <br />
<br />
* Every uplifting cardinal is a limit of $\Sigma_3$-reflecting cardinals, and is itself $\Sigma_3$-reflecting.<br />
* If $\kappa$ is the least uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
The analogous observation for pseudo uplifting cardinals holds as well, namely, every pseudo uplifting cardinal is $\Sigma_3$-reflecting and a limit of $\Sigma_3$-reflecting cardinals; and if $\kappa$ is the least pseudo uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.<br />
<br />
== Uplifting Laver functions ==<br />
<br />
Every uplifting cardinal admits an ordinal-anticipating Laver function, and indeed, a HOD-anticipating Laver function, a function $\ell:\kappa\to V_\kappa$, definable in $V_\kappa$, such that for any set $x\in\text{HOD}$ and $\theta$, there is an inaccessible cardinal $\gamma$ above $\theta$ such that $V_\kappa\prec V_\gamma$, for which $\ell^*(\kappa)=x$, where $\ell^*$ is the corresponding function defined in $V_\gamma$. <br />
<br />
== Connection with the resurrection axioms ==<br />
<br />
Many instances of the (weak) resurrection axiom imply that $\frak{c}^V$ is an uplifting cardinal in $L$:<br />
* RA(all) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* RA(ccc) implies that ${\frak c}^V$ is uplifting in $L$.<br />
* wRA(countably closed)+$\neg$CH implies that ${\frak c}^V$ is uplifting in $L$.<br />
* Under $\neg$CH, the weak resurrection axioms for the classes of axiom-A forcing, proper forcing, semi-proper forcing, and posets that preserve stationary subsets of $\omega_1$, respectively, each imply that ${\frak c}^V$ is uplifting in $L$.<br />
<br />
Conversely, if $\kappa$ is uplifting, then various resurrection axioms hold in a corresponding lottery-iteration forcing extension. <br />
<br />
'''Theorem.''' (Hamkins and Johnstone) The following theories are equiconsistent over ZFC:<br />
* There is an uplifting cardinal.<br />
* RA(all)<br />
* RA(ccc)<br />
* RA(semiproper)+$\neg$CH<br />
* RA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, RA($\alpha$-proper)+$\neg$CH<br />
* RA(axiom-A)+$\neg$CH<br />
* wRA(semiproper)+$\neg$CH<br />
* wRA(proper)+$\neg$CH<br />
* for some countable ordinal $\alpha$, wRA($\alpha$-proper})+$\neg$CH<br />
* wRA(axiom-A)+$\neg$CH<br />
* wRA(countably closed)+$\neg$CH<br />
<br />
{{References}}</div>Jdhhttp://cantorsattic.info/index.php?title=Pseudo_uplifting&diff=1114Pseudo uplifting2013-07-23T18:24:22Z<p>Jdh: Redirected page to Uplifting</p>
<hr />
<div>#REDIRECT [[uplifting]]</div>Jdhhttp://cantorsattic.info/index.php?title=Middle_attic&diff=1113Middle attic2013-07-23T18:22:13Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The middle attic}}<br />
[[File:StAugustineLighthouse.jpg | thumb | St. Augustine Lighthouse photo by Madrigar]]<br />
<br />
Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.<br />
<br />
* into the [[upper attic]]<br />
* [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals<br />
* [[extendible#-extendible cardinals | 0-extendible]] cardinal<br />
* [[extendible#$\Sigma_n$-extendible cardinals | $\Sigma_n$-extendible ]] cardinal<br />
* [[beth#beth_fixed_point | $\beth$-fixed point]]<br />
* the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]]<br />
* [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals<br />
* [[Theta | $\Theta$]]<br />
* the [[continuum]]<br />
* [[cardinal characteristics]] of the continuum<br />
** the [[bounding number | bounding number $\frak{b}$]], the [[dominating number | dominating number $\frak{d}$]], the [[covering number | covering numbers]], [[additivity number | additivity numbers]] and many more<br />
* the [[descriptive set theory | descriptive set-theoretic]] cardinals<br />
* [[aleph#aleph fixed point | $\aleph$-fixed point]]<br />
* the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]]<br />
* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals<br />
* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal<br />
* [[uncountable]], [[regular]] and [[successor]] cardinals<br />
* [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal<br />
* [[cardinal | cardinals]], [[infinite]] cardinals<br />
* [[Aleph zero | $\aleph_0$]] and the rest of the [[lower attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Middle_attic&diff=1112Middle attic2013-07-23T18:21:35Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The middle attic}}<br />
[[File:StAugustineLighthouse.jpg | thumb | St. Augustine Lighthouse photo by Madrigar]]<br />
<br />
Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.<br />
<br />
* into the [[upper attic]]<br />
* [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals<br />
* [[extendiable#-extendible cardinals | 0-extendible]] cardinal<br />
* [[extendible#$\Sigma_n$-extendible cardinals | $\Sigma_n$-extendible ]] cardinal<br />
* [[beth#beth_fixed_point | $\beth$-fixed point]]<br />
* the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]]<br />
* [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals<br />
* [[Theta | $\Theta$]]<br />
* the [[continuum]]<br />
* [[cardinal characteristics]] of the continuum<br />
** the [[bounding number | bounding number $\frak{b}$]], the [[dominating number | dominating number $\frak{d}$]], the [[covering number | covering numbers]], [[additivity number | additivity numbers]] and many more<br />
* the [[descriptive set theory | descriptive set-theoretic]] cardinals<br />
* [[aleph#aleph fixed point | $\aleph$-fixed point]]<br />
* the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]]<br />
* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals<br />
* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal<br />
* [[uncountable]], [[regular]] and [[successor]] cardinals<br />
* [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal<br />
* [[cardinal | cardinals]], [[infinite]] cardinals<br />
* [[Aleph zero | $\aleph_0$]] and the rest of the [[lower attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Middle_attic&diff=1111Middle attic2013-07-23T18:19:11Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The middle attic}}<br />
[[File:StAugustineLighthouse.jpg | thumb | St. Augustine Lighthouse photo by Madrigar]]<br />
<br />
Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.<br />
<br />
* into the [[upper attic]]<br />
* [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]]<br />
* [[reflecting#$\Sigma_2$ correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals<br />
* [[0-extendible]] cardinal<br />
* [[extendible#$\Sigma_n$-extendible cardinals | $\Sigma_n$-extendible ]] cardinal<br />
* [[beth#beth_fixed_point | $\beth$-fixed point]]<br />
* the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]]<br />
* [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals<br />
* [[Theta | $\Theta$]]<br />
* the [[continuum]]<br />
* [[cardinal characteristics]] of the continuum<br />
** the [[bounding number | bounding number $\frak{b}$]], the [[dominating number | dominating number $\frak{d}$]], the [[covering number | covering numbers]], [[additivity number | additivity numbers]] and many more<br />
* the [[descriptive set theory | descriptive set-theoretic]] cardinals<br />
* [[aleph#aleph fixed point | $\aleph$-fixed point]]<br />
* the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]]<br />
* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals<br />
* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal<br />
* [[uncountable]], [[regular]] and [[successor]] cardinals<br />
* [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal<br />
* [[cardinal | cardinals]], [[infinite]] cardinals<br />
* [[Aleph zero | $\aleph_0$]] and the rest of the [[lower attic]]</div>Jdhhttp://cantorsattic.info/index.php?title=Extendible&diff=1110Extendible2013-07-23T18:07:23Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: Extendible cardinal}}<br />
<br />
<br />
A cardinal $\kappa$ is ''$\eta$-extendible'' for an ordinal $\eta$ if and only if there is an elementary embedding $j:V_{\kappa+\eta}\to V_\theta$, with critical point $\kappa$, for some ordinal $\theta$. The cardinal $\kappa$ is ''extendible'' if and only if it is $\eta$-extendible for every ordinal $\eta$. <br />
<br />
Extendibility is connected in strength with supercompactness. Every extendible cardinal is supercompact, since from the embeddings $j:V_\lambda\to V_\theta$ we may extract the induced supercompactness measures $X\in\mu\iff j''\delta\in j(X)$ for $X\subset P_\kappa\delta$, provided that $j(\kappa)\gt\delta$ and $P_\kappa\delta\subset V_\lambda$, which one can arrange. On the other hand, if $\kappa$ is $\theta$-supercompact, witnessed by $j:V\to M$, then $\kappa$ is $\delta$-extendible inside $M$, provided $\beth_\delta\leq\theta$, since the restricted elementary embedding $j\upharpoonright V_\delta:V_\delta\to j(V_\delta)=M_{j(\delta)}$ has size at most $\theta$ and is therefore in $M$, witnessing $\delta$-extendibility there. <br />
<br />
== 0-extendible cardinals == <br />
<br />
For the special case $\eta=0$, an inaccessible cardinal $\kappa$ is $0$-extendible if $V_\kappa\prec V_\theta$ for some ordinal $\theta$. By itself, this is a weak notion; it is just the first step toward $\kappa$ being a [[pseudo uplifting]] cardinal. In particular, the existence of a $0$-extendible cardinal is weaker than the existence of a [[Mahlo]] cardinal.<br />
<br />
== $(\Sigma_n,\eta)$-extendible cardinals ==<br />
<br />
A cardinal $\kappa$ is $(\Sigma_n,\eta)$-extendible, if there is a $\Sigma_n$-elementary embedding $j:V_{\kappa+\eta}\to V_\theta$ with critical point $\kappa$, for some ordinal $\theta$. These cardinals were introduced by Bagaria, Hamkins, Tsaprounis and Usuba <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite>. <br />
<br />
== $\Sigma_n$-extendible cardinals == <br />
<br />
The special case of $\eta=0$ leads to a much weaker notion. Specifically, a cardinal $\kappa$ is ''$\Sigma_n$-extendible'' if it is $(\Sigma_n,0)$-extendible, or more simply, if $V_\kappa\prec V_\theta$ for some ordinal $\theta$. Note that this does not necessarily imply that $\kappa$ is inaccessible, and indeed the existence of $\Sigma_n$-extendible cardinals is provable in ZFC via the reflection theorem. For example, every [[reflecting#Reflection and correctness | $\Sigma_n$ correct]] cardinal is $\Sigma_n$-extendible, since from $V_\kappa\prec_{\Sigma_n} V$ and $V_\lambda\prec_{\Sigma_n} V$, where $\kappa\lt\lambda$, it follows that $V_\kappa\prec_{\Sigma_n} V_\lambda$. So in fact there is a closed unbounded class of $\Sigma_n$-extendible cardinals. <br />
<br />
Similarly, every Mahlo cardinal $\kappa$ has a stationary set of inaccessible $\Sigma_n$-extendible cardinals $\gamma<\kappa$. <br />
<br />
{{references}}</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1109Library2013-07-23T18:04:44Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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<br />
Mathias, A. R. D. and Rosicky, Jirí},<br />
TITLE = "Definable orthogonality classes in accessible categories are<br />
small",<br />
NOTE = "submitted for publication",<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;<br />
dedicated to P. Erd&#337;s on his 60th birthday), Vol.<br />
I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency, <br />
author = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman},<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic},<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<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 />
#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
EDITION = {Third},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1108Library2013-07-23T17:57:51Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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<br />
Mathias, A. R. D. and Rosicky, Jirí},<br />
TITLE = "Definable orthogonality classes in accessible categories are<br />
small",<br />
NOTE = "submitted for publication",<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;<br />
dedicated to P. Erd&#337;s on his 60th birthday), Vol.<br />
I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@ARTICLE{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency <br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic],<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url = {http://www.sciencedirect.com/science/article/pii/S0168007212000966},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<br />
AUTHOR = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman}<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 />
#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
EDITION = {Third},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdhhttp://cantorsattic.info/index.php?title=Library&diff=1107Library2013-07-23T17:55:02Z<p>Jdh: </p>
<hr />
<div>{{DISPLAYTITLE: The Cantor's attic library}}<br />
<br />
[[file:StepUpTheLadderTowardsWisdomBySigfridLundberg.jpg | thumb | right | Step up the ladder towards wisdom, photo by Sigfrid Lundberg]]<br />
<br />
Welcome to the library, our central repository for references cited here on Cantor's attic. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
#AbramsonHarringtonKleinbergZwicker1977:FlippingProperties bibtex=@article {AbramsonHarringtonKleinbergZwicker1977:FlippingProperties,<br />
AUTHOR = {Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and<br />
Zwicker, William},<br />
TITLE = {Flipping properties: a unifying thread in the theory of large<br />
cardinals},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {12},<br />
YEAR = {1977},<br />
NUMBER = {1},<br />
PAGES = {25--58},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02K35 (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<br />
Mathias, A. R. D. and Rosicky, Jirí},<br />
TITLE = "Definable orthogonality classes in accessible categories are<br />
small",<br />
NOTE = "submitted for publication",<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 />
<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;<br />
dedicated to P. Erd&#337;s on his 60th birthday), Vol.<br />
I},<br />
PAGES = {109--130. Colloq. Math. Soc. J&#225;nos Bolyai, Vol. 10},<br />
PUBLISHER = {North-Holland},<br />
ADDRESS = {Amsterdam},<br />
YEAR = {1975},<br />
MRCLASS = {02K35 (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 />
#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 />
#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 />
<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 />
<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. Drake},<br />
DOI = {10.1016/0003-4843(81)90011-5},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(81)90011-5},<br />
}<br />
<br />
#ErdosHajnal1962:Ramsey bibtex=@article {ErdosHajnal1962:Ramsey,<br />
AUTHOR = {Erd&#337;s, Paul and Hajnal, Andras},<br />
TITLE = {Some remarks concerning our paper ``On the structure of<br />
set-mappings''. Non-existence of a two-valued $\sigma<br />
$-measure for the first uncountable inaccessible cardinal},<br />
JOURNAL = {Acta Math. Acad. Sci. Hungar.},<br />
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},<br />
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&#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 />
#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 />
#Gaifman1974:ElementaryEmbeddings bibtex=@incollection{Gaifman1974:ElementaryEmbeddings,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Elementary embeddings of models of set-theory and certain<br />
subtheories},<br />
BOOKTITLE = {Axiomatic set theory (Proc. Sympos. Pure Math., Vol.<br />
XIII, Part II, Univ. California, Los Angeles,<br />
Calif., 1967)},<br />
PAGES = {33--101},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence R.I.},<br />
YEAR = {1974},<br />
MRCLASS = {02K15 (02H13)},<br />
MRNUMBER = {0376347 (51 \#12523)},<br />
MRREVIEWER = {L. Bukovsky},<br />
}<br />
<br />
#Gitman2011:RamseyLikeCardinals bibtex=@ARTICLE {Gitman2011:RamseyLikeCardinals,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Ramsey-like cardinals},<br />
JOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {519-540},<br />
MRNUMBER = {2830415},<br />
EPRINT={0801.4723},<br />
URL={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf}}<br />
<br />
#GitmanWelch2011:RamseyLikeCardinalsII bibtex=@article {GitmanWelch2011:RamseyLikeCardinalsII,<br />
AUTHOR = {Gitman, Victoria and Welch, Philip},<br />
TITLE = {Ramsey-like cardinals II},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {76},<br />
YEAR = {2011},<br />
NUMBER = {2},<br />
PAGES = {541--560},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03E55},<br />
MRNUMBER = {2830435},<br />
EPRINT ={1104.4448},<br />
URL = {http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},<br />
}<br />
#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. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {60},<br />
NUMBER = {1},<br />
YEAR = {1995},<br />
PAGES = {58--73},<br />
URL = {http://www.jstor.org/stable/2275509}<br />
}<br />
<br />
#HamkinsLewis2000:InfiniteTimeTM bibtex=@article {HamkinsLewis2000:InfiniteTimeTM,<br />
AUTHOR = {Hamkins, Joel David and Lewis, Andy},<br />
TITLE = {Infinite time Turing machines},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {65},<br />
YEAR = {2000},<br />
NUMBER = {2},<br />
PAGES = {567--604},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {1771072 (2001g:03072)},<br />
MRREVIEWER = {Robert M. Baer},<br />
DOI = {10.2307/2586556},<br />
URL = {http://dx.doi.org/10.2307/2586556},<br />
eprint = {math/9808093}<br />
}<br />
<br />
<br />
#Hamkins2002:Turing bibtex=@ARTICLE{Hamkins2002:Turing,<br />
author = {Hamkins, Joel David},<br />
title = {Infinite time Turing machines},<br />
journal = {Minds and Machines},<br />
year = {2002},<br />
volume = {12},<br />
number = {4},<br />
pages = {521--539},<br />
month = {},<br />
note = {special issue devoted to hypercomputation},<br />
key = {},<br />
annote = {},<br />
eprint = {math/0212047},<br />
url = {http://boolesrings.org/hamkins/turing-mm/},<br />
}<br />
<br />
#Hamkins2004:SupertaskComputation bibtex=@INPROCEEDINGS{Hamkins2004:SupertaskComputation,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {Supertask computation},<br />
BOOKTITLE = {Classical and new paradigms of computation and their complexity hierarchies},<br />
SERIES = {Trends Log. Stud. Log. Libr.},<br />
VOLUME = {23},<br />
PAGES = {141--158},<br />
PUBLISHER = {Kluwer Acad. Publ.},<br />
ADDRESS = {Dordrecht},<br />
YEAR = {2004},<br />
MRCLASS = {03D10 (03D25 68Q05)},<br />
MRNUMBER = {2155535},<br />
DOI = {10.1007/978-1-4020-2776-5_8},<br />
URL = {http://dx.doi.org/10.1007/978-1-4020-2776-5_8},<br />
note = {Papers of the conference ``Foundations of the Formal Sciences III'' held in Vienna, September 21-24, 2001},<br />
eprint = {math/0212049},<br />
file = F,<br />
}<br />
<br />
#Hamkins2001:WholenessAxiomAndVequalHOD bibtex=@article{Hamkins2001:WholenessAxiom,<br />
AUTHOR = {Hamkins, Joel David},<br />
TITLE = {The wholeness axioms and V=HOD},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {40},<br />
YEAR = {2001},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03E35 (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 />
#HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@ARTICLE{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency bibtex=@article{HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency,<br />
title = {Generalizations of the {Kunen} inconsistency},<br />
journal = {Annals of Pure and Applied Logic],<br />
volume = {163},<br />
number = {12},<br />
pages = {1872 - 1890},<br />
year = {2012},<br />
issn = {0168-0072},<br />
doi = {10.1016/j.apal.2012.06.001},<br />
eprint = {1106.1951},<br />
url = {http://www.sciencedirect.com/science/article/pii/S0168007212000966},<br />
url={http://jdh.hamkins.org/generalizationsofkuneninconsistency/},<br />
AUTHOR = {Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman}<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 />
#Jech2003:SetTheory bibtex=@book{Jech2003:SetTheory,<br />
AUTHOR = {Jech, Thomas J.},<br />
TITLE = {Set Theory},<br />
EDITION = {Third},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {2003},<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 = {http://math.bu.edu/people/aki/intro.pdf}<br />
}<br />
<br />
#Kanamori1978:StrongAxioms bibtex=@article <br />
{Kanamori1978:StrongAxioms, <br />
author = {Kanamori, Akihiro and Reinhardt, William N. and Solovay, Robert M.}, <br />
title = {Strong axioms of infinity and elementary embeddings}, <br />
note = {In ''Annals of Mathematical Logic'', '''13'''(1978)}, <br />
year = {1978}, <br />
url = {http://math.bu.edu/people/aki/d.pdf},}<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 />
#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 />
<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 />
#SharpeWelch2011:GreatlyErdosChang bibtex=@article {SharpeWelch2011:GreatlyErdosChang,<br />
AUTHOR = {Sharpe, Ian and Welch, Philip},<br />
TITLE = {Greatly Erd&#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 />
#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 />
<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 />
#Welch2000:Eventually bibtex=@article{Welch2000:Eventually,<br />
author = {Welch, Philip},<br />
title = {Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable reals},<br />
journal = {Journal of Symbolic Logic},<br />
volume = {65},<br />
year = {2000},<br />
number = {3},<br />
pages = {1193--1203},<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 />
</biblio><br />
<br />
== User instructions == <br />
<br />
Cantor's attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>Jdh