Supercompact Cardinals Research Articles

Supercompactness and partial level by level equivalence between strong compactness and strongness

We force and construct a model containing supercompact cardinals in which, for any measurable cardinal $\delta $ and any ordinal $\alpha $ below the least beth fixed point above $\delta $, if $\delta ^{+ \alpha }$ is regular, $\delta $ is $\delta ^{+ \alp

Failures of GCH and the level by level equivalence between strong compactness and supercompactness

AbstractWe force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Characterizing strong compactness via strongness

AbstractWe construct a model in which the strongly compact cardinals can be non‐trivially characterized via the statement “κ is strongly compact iff κ is a measurable limit of strong cardinals”. If our ground model contains large enough cardinals, there will be supercompact cardinals in the universe containing this characterization of the strongly compact cardinals.

Some structural results concerning supercompact cardinals

Abstract.We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ+ supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.

Strong compactness, measurability, and the class of supercompact cardinals

We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit o

Identity crises and strong compactness

From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal $\kappa$ is $2^\kappa = \kappa^+$ supercompact.

Martin's Maximum and the [formula omitted

Assuming the existence of a supercompact limit of supercompact cardinals, we modify the original consistency proof of Martin's Maximum to obtain a model in which MM holds but the P max axiom (∗) fails.

The lottery preparation

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 ⩽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⊆κ avoiding 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.

On measurable limits of compact cardinals

AbstractWe extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

Gap Forcing: Generalizing the Lévy-Solovay Theorem

AbstractThe Lévy-Solovay Theorem[8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

Laver indestructibility and the class of compact cardinals

AbstractUsing an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.

Destruction or preservation as you like it

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of < κ-directed closed posets which preserve a supercompact cardinal κ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible.

The Tree Property

We construct a model in which there are no ℵn-Aronszajn trees for any finiten⩾2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is noκ++-Aronszajn tree forκa strong limit cardinal of cofinalityω, starting from a model with a supercompact cardinal and a weakly compact cardinal above it.

Patterns of compact cardinals

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V ⊨ “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P ∈ V so that for V ̄ = V P , V ̄ Ω ⊨ “ZFC + There is a proper class of compact cardinals + If ƒ(α) = 0, then the αth compact cardinal is not supercompact + If ƒ(α) = 1, then the αth compact cardinal is supercompact”. We then prove a generalized version of this theorem assuming κ is a supercompact limit of supercompact cardinals and ƒ : κ → 2 is a function, and we derive as corollaries of the generalized version of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of nonsupercompact strongly compact cardinals and the consistency of the least supercompact cardinal being a limit of strongly compact cardinals.

More on the Least Strongly Compact Cardinal

AbstractWe 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 &gt; 2k supercompact.

Consistency strength of the axiom of full reflection at large cardinals

We prove that the Axiom of Full Reflection at a measurable cardinal is equiconsistent with the existence of a measurable cardinal. We generalize the result also to larger cardinals such as strong or supercompact cardinals.

The two-cardinals transfer property and resurrection of supercompactness 󠀼span style=󠀢color:red󠀢󠀾This article has been retracted󠀼/span󠀾

We show that the transfer property ( ℵ 1 , ℵ 0 ) → ( λ + , λ ) (\aleph _1,\aleph _0)\to (\lambda ^+,\lambda ) for singular λ \lambda does not imply (even) the existence of a non-reflecting stationary subset of λ + \lambda ^+ . The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.

On the cardinality of Lindelöf spaces with points Gδ

We investigate Arhangel'skǐi's problem of whether Lindelöf T 2 spaces with points G δ have cardinality ⩽ 2 ℵ 0 . We extend results of Shelah, using supercompact cardinals to exclude cardinalities between ℵ 1 and 2 ℵ 1 . We also obtain several results concerning the possible structure of large Lindelöf spaces with points G δ .

Donald A. Martin and John R. Steel. Projective determinacy. Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), pp. 6582–6586. - W. Hugh Woodin. Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), pp. 6587–6591. - Donald

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A note on a result of Kunen and Pelletier

AbstractSuppose that U and U′ are normal ultrafilters associated with some supercompact cardinal. How may we compare U and U′? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study.In [6], Menas introduced a combinatorial principle χ(U) of normal ultrafilters U associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for U does not imply χ(U). Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy χ(U). Our method yields a large collection of such normal ultrafilters.