Universal indestructibility for degrees of supercompactness and strongly compact cardinals

We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness. In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly compact cardinals, each of which exhibits a significant amount of indestructibility for its strong compactness. The first author’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. The first author wishes to thank James Cummings for helpful discussions on the subject matter of this paper. In addition, both authors wish to thank the referee, for many helpful comments and suggestions which were incorporated into the current version of the paper.     

Supercompactness and level by level equivalence are compatible with indestructibility for strong compactness

It is known that if are such that κ is indestructibly supercompact and λ is 2^λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible under κ-directed closed forcing. The author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.     

Universal partial indestructibility and strong compactness

For any ordinal δ, let λ_δ be the least inaccessible cardinal above δ. We force and construct a model in which the least supercompact cardinal κ is indestructible under κ‐directed closed forcing and in which every measurable cardinal δ < κ is < λ_δ strongly compact and has its < λ_δ strong compactness indestructible under δ‐directed closed forcing of rank less than λ_δ. In this model, κ is also the least strongly compact cardinal. We also establish versions of this result in which κ is the least strongly compact cardinal but is not supercompact. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some remarks on indestructibility and Hamkins’ lottery preparation

In this paper, we first prove several general theorems about strongness, supercompactness, and indestructibility, along the way giving some new applications of Hamkins lottery preparation forcing to indestructibility. We then show that it is consistent, relative to the existence of cardinals < so that is supercompact and is inaccessible, for the least strongly compact cardinal to be the least strong cardinal and to have its strongness, but not its strong compactness, indestructible under -strategically closed forcing. Mathematics Subject Classification (2000):03E35, 03E55     

Reducing the consistency strength of an indestructibility theorem

Using 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 & Co. KGaA, Weinheim)     

Strong Compactness and a Global Version of a Theorem of Ben‐David and Magidor

Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ⁺ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □_γ+ω fails but □_{γ+ω, ω} holds. This generalizes a result of Ben‐David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.     

More on the Least Strongly Compact Cardinal

We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2^k supercompact.     

On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact', then is supercompact' and so that, (b) (equivalence) for regular, is strongly compact' iff is supercompact', except possibly if is a measurable limit of cardinals which are supercompact.

     

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. The first author's research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.     

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, holds for every regular uncountable cardinal , and below the least supercompact cardinal , holds on a stationary subset of . There are no restrictions in our model on the structure of the class of supercompact cardinals.The author wishes to thank the referee for numerous helpful comments and suggestions, which have considerably improved the presentation of the material contained herein. The author also wishes to thank Andreas Blass, the corresponding editor, for a useful suggestion, and Grigor Sargsyan for a very helpful conversation on the subject matter of this paper.Mathematics Subject Classification (2000): 03E35, 03E55     

Large cardinals with few measures

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

     

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.     

Large Cardinals and Ramifiability for Directed Sets

The notion of “ramifiability” (or “tree‐property”), usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.     

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω₂. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.     

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

We 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, 2^δ > δ ⁺ and δ is < 2^δ 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 & Co. KGaA, Weinheim)     

Forcing the Least Measurable to Violate GCH

Starting with a model for “GCH + k is k⁺ supercompact”, we force and construct a model for “k is the least measurable cardinal + 2^k = K⁺”. This model has the property that forcing over it with Add(k,k⁺⁺) preserves the fact k is the least measurable cardinal.