Inner models with large cardinal features usually obtained by forcing

Posted on November 3, 2011 by Joel David Hamkins

[bibtex key=ApterGitmanHamkins2012:InnerModelsWithLargeCardinals]

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal $κ$ for which $2^{κ} = κ^{+}$ , another for which $2^{κ} = κ^{+ +}$ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model $W$ with a strongly compact cardinal $κ$ , such that $H_{κ^{+}}^{V} \subseteq H O D^{W}$ . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit $δ$ of $< δ$ -supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

What is the theory ZFC without power set?

Posted on October 13, 2011 by Joel David Hamkins

[bibtex key=”GitmanHamkinsJohnstone2016:WhatIsTheTheoryZFC-Powerset?”]

This is joint work with Victoria Gitman and Thomas Johnstone.

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC- in which $ω_{1}$ is singular, in which every set of reals is countable, yet $ω_{1}$ exists, in which there are sets of reals of every size $ℵ_{n}$ , but none of size $ℵ_{ω}$ , and therefore, in which the collection axiom sceme fails; there are models of ZFC- for which the Los theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC- for which the Gaifman theorem fails, in that there is an embedding $j : M \to N$ of ZFC- models that is $Σ_{1}$ -elementary and cofinal, but not elementary; there are elementary embeddings $j : M \to N$ of ZFC- models whose cofinal restriction $j : M \to ⋃ j “ M$ is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a $Σ_{1}$ -formula or a $Π_{1}$ -formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ${ZFC}^{-}$ , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.

See Victoria Gitman’s summary post on the article

Generalizations of the Kunen inconsistency, KGRC, Vienna 2011

Posted on September 27, 2011 by Joel David Hamkins

This is a talk at the research seminar of the Kurt Gödel Research Center, November 3, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others. For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V. Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice. I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions. This is joint work with Greg Kirmayer and Norman Perlmutter.

Slides | Article

Generalizations of the Kunen inconsistency

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsKirmayerPerlmutter2012:GeneralizationsOfKunenInconsistency]

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.

Indestructible strong unfoldability

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsJohnstone2010:IndestructibleStrongUnfoldability]

Using the lottery preparation, we prove that any strongly unfoldable cardinal $κ$ can be made indestructible by all $< κ$ -closed + $κ^{+}$ -preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of $< κ$ -closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $κ$ can be made indestructible by all $< κ$ -closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by $< κ$ -directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.

Tall cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2009:TallCardinals]

A cardinal $κ$ is tall if for every ordinal $θ$ there is an embedding $j : V \to M$ with critical point $κ$ such that $j (κ) > θ$ and $M^{κ} \subset M$ . Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal $κ$ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of $2^{κ}$ as high as desired.

The proper and semi-proper forcing axioms for forcing notions that preserve $ℵ_{2}$ or $ℵ_{3}$

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsJohnstone2009:PFA(aleph_2-preserving)]

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $κ$ under $\neg 0^{♯}$ forces $PFA (ℵ_{2} -preserving)$ , $PFA (ℵ_{3} -preserving)$ and ${PFA}_{ℵ_{2}}$ , with $2^{ω} = κ = ℵ_{2}$ . The method adapts to semi-proper forcing, giving $SPFA (ℵ_{2} -preserving)$ , $SPFA (ℵ_{3} -preserving)$ and ${SPFA}_{ℵ_{2}}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $SPFA (ℵ_{2} -preserving) + SPFA (ℵ_{3} -preserving) + {SPFA}_{ℵ_{2}} + 2^{ω} = ℵ_{2}$ . Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $ℵ_{3}$ to $ℵ_{1}$ .

Large cardinals with few measures

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures]

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $κ^{+}$ many normal measures on the least measurable cardinal $κ$ . This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $λ$ -strong compactness or $λ$ -supercompactness measures on $P_{κ} (λ)$ can be exactly $λ^{+}$ , if $λ > κ$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

Extensions with the approximation and cover properties have no new large cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2003:ExtensionsWithApproximationAndCoverProperties]

If an extension $\bar{V}$ of $V$ satisfies the $δ$ -approximation and cover properties for classes and $V$ is a class in $\bar{V}$ , then every suitably closed embedding $j : \bar{V} \to \bar{N}$ in $\bar{V}$ with critical point above $δ$ restricts to an embedding $j ↾ V : V \to N$ amenable to the ground model $V$ . In such extensions, therefore, there are no new large cardinals above delta. This result extends work in my article on gap forcing.

Exactly controlling the non-supercompact strongly compact cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterHamkins2003:ExactlyControlling]

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

A simple maximality principle

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2003:MaximalityPrinciple]

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence $φ$ holding in some forcing extension $V^{P}$ and all subsequent extensions $V^{P * Q}$ holds already in $V$ . It follows, in fact, that such sentences must also hold in all forcing extensions of $V$ . In modal terms, therefore, the Maximality Principle is expressed by the scheme $(◊ φ) \to φ$ , and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $φ$ , is equiconsistent with the scheme asserting that $V_{δ}$ is an elementary substructure of $V$ for an inaccessible cardinal $δ$ , which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0^{♯}$ exists, that $x^{♯}$ exists for every set $x$ , that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

Indestructibility and the level-by-level agreement between strong compactness and supercompactness

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterHamkins2002:LevelByLevel]

Can a supercompact cardinal $κ$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $κ$ , then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.

Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterHamkins2001:IndestructibleWC]

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

The wholeness axioms and $V = H O D$

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2001:WholenessAxiom]

The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding $j : V \to V$ . Formalized by augmenting the usual language of set theory with an additional unary function symbol j to represent the embedding, they avoid the Kunen inconsistency by restricting the base theory ZFC to the usual language of set theory. Thus, under the Wholeness Axioms one cannot appeal to the Replacement Axiom in the language with j as Kunen does in his famous inconsistency proof. Indeed, it is easy to see that the Wholeness Axioms have a consistency strength strictly below the existence of an $I_{3}$ cardinal. In this paper, I prove that if the Wholeness Axiom $W A_{0}$ is itself consistent, then it is consistent with $V = H O D$ . A consequence of the proof is that the various Wholeness Axioms $W A_{n}$ are not all equivalent. Furthermore, the theory $Z F C + W A_{0}$ is finitely axiomatizable.

The lottery preparation

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2000:LotteryPreparation]

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal $κ$ , for example, becomes fully indestructible by $κ$ -directed closed forcing; a strong cardinal $κ$ becomes indestructible by less-than-or-equal- $κ$ -strategically closed forcing; and a strongly compact cardinal $κ$ becomes indestructible by, among others, the forcing to add a Cohen subset to $κ$ , the forcing to shoot a club $C$ in $κ$ which avoids the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context.

Joel David Hamkins

mathematics and philosophy of the infinite

Tag Archives: large cardinals

Inner models with large cardinal features usually obtained by forcing

What is the theory ZFC without power set?

Generalizations of the Kunen inconsistency, KGRC, Vienna 2011

Generalizations of the Kunen inconsistency

Indestructible strong unfoldability

Tall cardinals

The proper and semi-proper forcing axioms for forcing notions that preserve $ℵ_{2}$ or $ℵ_{3}$

Large cardinals with few measures

Extensions with the approximation and cover properties have no new large cardinals

Exactly controlling the non-supercompact strongly compact cardinals

Indestructibility and the level-by-level agreement between strong compactness and supercompactness

Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

The wholeness axioms and $V = H O D$

The lottery preparation

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