Supercompact Cardinals Research Articles

STATIONARY REFLECTION AND THE FAILURE OF THE SCH

AbstractIn this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $ , this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\operatorname {\mathrm {cf}}(\nu ) = \omega $ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

Sigma-Prikry forcing II: Iteration Scheme

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a non-reflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

Strong compactness and the ultrapower axiom I: the least strongly compact cardinal

The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.

INDESTRUCTIBILITY WHEN THE FIRST TWO MEASURABLE CARDINALS ARE STRONGLY COMPACT

AbstractWe prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$ , we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible (i.e., $\kappa _1$ ’s strong compactness is indestructible under arbitrary $\kappa _1$ -directed closed forcing), and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $ . This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.

Strongly compact cardinals and the continuum function

We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals. We begin by showing that it is possible to force violations of GCH at an arbitrary strongly compact cardinal using only strong compactness as our initial assumption. This result is due to the third author. We then investigate realising Easton functions at and above the least measurable limit of supercompact cardinals starting from an initial assumption of the existence of a measurable limit of supercompact cardinals. By results due to Menas, assuming 2κ=κ+, the least measurable limit of supercompact cardinals κ is provably in ZFC a non-supercompact strongly compact cardinal which is not κ+-supercompact. We also consider generalisations of our earlier theorems in the presence of more than one strongly compact cardinal. We conclude with some open questions.

Easton's theorem for the tree property below ℵω

Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal ℵn, 1<n<ω, is consistent with an arbitrary continuum function below ℵω which satisfies 2ℵn>ℵn+1, n<ω. Thus the tree property has no provable effect on the continuum function below ℵω except for the restriction that the tree property at κ++ implies 2κ>κ+ for every infinite κ.

On the relationship between mutual and tight stationarity

We construct a model where every increasing ω-sequence of regular cardinals carries a mutually stationary sequence which is not tightly stationary, and show that this property is preserved under a class of Prikry-type forcings. Along the way, we give examples in the Cohen and Prikry models of ω-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality ω1 ordinals, and show that such stationary sequences are mutually stationary in the presence of interleaved supercompact cardinals.

More on HOD-supercompactness

We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact.

Guessing models and the approachability ideal

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

Sigma-Prikry forcing I: The Axioms

AbstractWe introduce a class of notions of forcing which we call$\Sigma $-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are$\Sigma $-Prikry. We show that given a$\Sigma $-Prikry poset$\mathbb P$and a name for a non-reflecting stationary setT, there exists a corresponding$\Sigma $-Prikry poset that projects to$\mathbb P$and kills the stationarity ofT. Then, in a sequel to this paper, we develop an iteration scheme for$\Sigma $-Prikry posets. Putting the two works together, we obtain a proof of the following.Theorem.If$\kappa $is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which$\kappa $remains a strong limit cardinal, every finite collection of stationary subsets of$\kappa ^+$reflects simultaneously, and$2^\kappa =\kappa ^{++}$.

The super tree property at the successor of a singular

For an inaccessible cardinal κ, the super tree property (ITP) at κ holds if and only if κ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of ω many supercompact cardinals. Then we show that it can consistently hold at ℵω+1. We also consider a stronger principle, ISP and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining large cardinal-type properties at smaller cardinals.

The special Aronszajn tree property

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.

WEAK SQUARES AND VERY GOOD SCALES

AbstractWe assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where □K,&lt;K holds but □K, λ fails for λ &lt; K.

The tree property on a countable segment of successors of singular cardinals

Starting from the existence of many supercompact cardinals, we construct a model of ${\rm ZFC}+{\rm GCH}$ in which the tree property holds at a countable segment of successors of singular cardinals.

The Hypothesis and a supercompact cardinal

AbstractIn this paper, we prove that: if κ is supercompact and the Hypothesis holds, then there is a proper class of regular cardinals in which are measurable in . Woodin also proved this result independently . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the Hypothesis and supercompact cardinals, large cardinals in are reflected to be large cardinals in in a local way, and reveals the huge difference between ‐supercompact cardinals and supercompact cardinals under the Hypothesis.

Absoluteness via resurrection

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

The tree property at the successor of a singular limit of measurable cardinals

Assume $$\lambda $$ is a singular limit of $$\eta $$ supercompact cardinals, where $$\eta \le \lambda $$ is a limit ordinal. We present two methods for arranging the tree property to hold at $$\lambda ^{+}$$ while making $$\lambda ^{+}$$ the successor of the limit of the first $$\eta $$ measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at $$\aleph _{\eta ^2+1}$$ with the failure of SCH at $$\aleph _{\eta ^2}$$ . This extends results of Neeman and Sinapova. The second method is also used to get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova.

Morasses, semimorasses and supercompact ultrafilters

We prove that morasses are not only compatible with supercompact cardinals, but, moreover, simplified morasses can be elements of the supercompact ultrafilter. We also give an application of this fact to the construction of continuous tree-like scales.

REFLECTION OF STATIONARY SETS AND THE TREE PROPERTY AT THE SUCCESSOR OF A SINGULAR CARDINAL

AbstractWe show that from infinitely many supercompact cardinals one can force a model of ZFC where both the tree property and the stationary reflection hold at אω2+1.

The strong tree property and weak square

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at for all but there are weak square and a very good scale at .