Supercompact Cardinals Research Articles

The tree property at ℵω+1

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1· This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor–Shelah [7].

More Easton theorems for level by level equivalence

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize [1, Theorem 1]. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.

The Wholeness Axioms and the Class of Supercompact Cardinals

We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

Inner models with large cardinal features usually obtained by forcing

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 2κ = κ +, another for which 2κ = 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 \({H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}\). 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.

A remark on the tree property in a choiceless context

We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ $${{\rm ZF} + \neg{\rm AC}_\omega}$$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from hypotheses stronger in consistency strength than a supercompact limit of supercompact cardinals. A lower bound in consistency strength is provided by a result of Busche and Schindler, who showed that the consistency of the theory “ZF + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” implies the consistency of ADL(R).

Coding into HOD via normal measures with some applications

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

A cofinality-preserving small forcing may introduce a special Aronszajn tree

It is relatively consistent with the existence of two supercompact cardinals that a special Aronszajn tree of height \({\aleph_{\omega_1+1}}\) is introduced by a cofinality-preserving forcing of size \({\aleph_3}\).

Epireflections and supercompact cardinals

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → L X is an epimorphism for all X and the class of L -local objects is defined by an absolute formula, then the existence of a sufficiently large supercompact cardinal implies that L is a localization with respect to some set of morphisms.

Large cardinals and locally defined well-orders of the universe

By forcing over a model of ZFC + GCH (above ℵ 0 ) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H ( κ + ) ( κ ≥ ω 2 a regular cardinal) is a well-order of H ( κ + ) definable over the structure 〈 H ( κ + ) , ∈ 〉 by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n -huge cardinals from the ground model (for all n ).

Martin’s Maximum and definability in [formula omitted

In [P. Larson, Martin’s Maximum and the P max axiom (∗), Ann. Pure App. Logic 106 (1–3) (2000) 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin’s Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 (1988) 1–47] to show that from a supercompact limit of supercompact cardinals one could force Martin’s Maximum to hold while the P max axiom (∗) fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H ( ℵ 2 ) without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result.

An L-like model containing very large cardinals

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.

Reducing the consistency strength of an indestructibility theorem

AbstractUsing an idea of Sargsyan, we show how to reduce the consistency strength of the assumptions employed to establish a theorem concerning a uniform level of indestructibility for both strong and supercompact cardinals. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

The ground axiom

AbstractA new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.

Supercompactness and measurable limits of strong cardinals II: Applications to level by level equivalence

AbstractWe construct models for the level by level equivalence between strong compactness and supercompactness in which for κ the least supercompact cardinal and δ ≤ κ any cardinal which is either a strong cardinal or a measurable limit of strong cardinals, 2δ &gt; δ + and δ is &lt; 2δ supercompact. In these models, the structure of the class of supercompact cardinals can be arbitrary, and the size of the power set of κ can essentially be made as large as desired. This extends and generalizes [5, Theorem 2] and [4, Theorem 4]. We also sketch how our techniques can be used to establish a weak indestructibility result. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Failures of SCH and Level by Level Equivalence

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.

Identity crises and strong compactness III: Woodin cardinals

We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the n th strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships between Woodin cardinals and strongly compact cardinals.

An Easton theorem for level by level equivalence

We establish an Easton theorem for the least supercompact cardinal that is consistent with the level by level equivalence between strong compactness and supercompactness. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals. We also briefly indicate how our methods of proof yield an Easton theorem that is consistent with the level by level equivalence between strong compactness and supercompactness in a universe with a restricted number of large cardinals. We conclude by posing some related open questions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

More on regular reduced products

Abstract.The authors show, by means of a finitary version of the combinatorial principle of [7]. the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ. if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ+ on ΠiMi/D and ΠiNi/D. If in addition 2λ = λ+ and i &lt; λ implies |Mi| + |Ni| ≤ λ+ this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in [2].

Diamond, square, and level by level equivalence

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.