The main result of this paper, built on previous work by the author and T. Wilson, is the proof that the theory “ADR+DC + there is an R-complete measure on Θ” is equiconsistent with “ZF+DC+ ADR + there is a supercompact measure on ℘ω1(℘(R))+Θ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of ZF+DC.

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j:V→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j: V \to M}$$\end{document} such that M is closed under sequences of length sup{j|f:κ→κ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sup\{{j\,|\,f: \kappa \to \kappa}\}}$$\end{document}. Some of the (...) other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Among the results, I highlight the following. Vopěnka cardinals are the same as Woodin-for-supercompactness cardinals.There are no excessively hypercompact cardinals.Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. (shrink)

We show that all the projective ordinals $\delta^1_n$ are supercompact through their supremum $\aleph_{\varepsilon 0}$, and a ways beyond.

Motivated by the goal of constructing a model in which there are no κ-Aronszajn trees for any regular $k>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.

Given a cardinal κ that is λ-supercompact for some regular cardinal λ⩾κ and assuming GCH, we show that one can force the continuum function to agree with any function F:[κ,λ]∩REG→CARD satisfying ∀α,β∈domα F. Our argument extends Woodinʼs technique of surgically modifying a generic filter to a new case: Woodinʼs key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on (...) the ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions. (shrink)

During his Fall 2005 set theory seminar, Woodin asked whether V-supercompactness implies HOD-supercompactness. We show, as he predicted, that that the answer is no.

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal. Utilizing the failure of a weak version of a square, we show that the best currently known lower bounds for the consistency strength of these principles can be applied.

The extender based Prikry forcing notion is being generalized to super compact extenders.

It is known that if $\kappa < \lambda$ are such that κ is indestructibly supercompact and λ is 2λ 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. (shrink)

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, 2δ > δ+ and δ is < 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. (shrink)

A cardinal κ is nearly θ-supercompact if for every A⊆θ, there exists a transitive M⊨ZFC− closed under θ and j″θ∈N.2 This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly 2θ<κ-supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that θ<κ=θ, we can move to a forcing extension preserving all cardinals below θ++ where κ remains θ-supercompact but is not (...) nearly θ+-supercompact. We will also show that if κ is nearly θ-supercompact for some θ⩾2κ such that θ<θ=θ, then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result. A forcing poset is <κ-directed closed if it is γ-directed closed for all γ<κ in the sense of Jech [13, Def. 21.6]. We will prove that if κ is nearly θ-supercompact for some θ⩾κ such that θ<θ=θ, then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, we will show that if κ is nearly θ-supercompact for some θ⩾κ+ for which θ<θ=θ, then AD holds in L. In particular, if κ is nearly κ+-supercompact and 2κ=κ+, then AD holds in L. (shrink)

We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study Cn‐supercompactness.

In this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.

In this paper we study the notion of $C^{}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{}$ -supercompact but also that the least supercompact is $C^{}$ -supercompact }$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

The main result of this paper is Theorem, which shows that it is possible for derived models to satisfy “ω1 is ‐supercompact”. Other constructions of models of this theory are also discussed; in particular, Theorem constructs a normal fine measure on and hence a model of “Θ is regular”+“ω1 is ‐supercompact” from a model of “Θ is measurable”.

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 ↠ 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. (shrink)

We give a new characterization of λ-supercompact cardinal κ in terms of -Solovay pairs. We give some applications of -Solovay pairs.

Suppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on P κ (λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

Working in ZF + DC with no additional use of the axiom of choice, we show how to iterate the extended ultrapower construction of Spector . This generalizes the technique of iterated ultrapowers to choiceless set theory. As an application, we prove the following theorem: Assume V = LU[κ] + “κ is λ-supercompact with normal ultrafilter U” + DC. Then for every sufficiently large regular cardinal ρ, there exists a set-generic extension V[G] of the universe in which there exists (...) for some σ a set S ρ for which one can define an elementary embedding j mapping V to LD[S], where D is the filter in V[G] generated by the closed unbounded filter on ρ. Moreover, we have j = ρ, j = σ, j) = S according to LD[S]), and j = D ∩ LD[S] i s a normal ultrafilter in LD[S] on ρ. (shrink)

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every Π11 sentence true in a structure M (of any size) containing (...) κ in a language of size less than κ is also true in a substructure m≺M of size less than κ with m∩κ∈κ. (shrink)

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.

We show how the use of a Laver function in the proof of the consistency, relative to the existence of a supercompact cardinal, of both the Proper Forcing Axiom and the Semiproper Forcing Axiom can be eliminated via the use of lottery sums of the appropriate partial orderings.

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. (shrink)

We characterize some large cardinal properties, such as $\mu$-measurability and $P^2(\kappa)$-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on $P_\kappa(2^\kappa)$. This leads to the characterization of $2^\kappa$-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, $\mathrm{Full}_\kappa$, of $P_\kappa(2^\kappa)$, whose elements code measures on cardinals less than $\kappa$. The hypothesis that $\mathrm{Full}_\kappa$ is stationary (a weaker assumption than $2^\kappa$-supercompactness) is equivalent (...) to a higher order Lowenheim-Skolem property, and settles a question about directed versus chain-type unions on $P_\kappa\lambda$. (shrink)

T. K. Menas [4, pp. 225-234] introduced a combinatorial property χ (μ) of a measure μ on a supercompact cardinal κ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property. We also show that if α is the least cardinal greater than κ such that P κ α bears a measure without the partition property, then α is inaccessible and Π 2 1 -indescribable.

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 Theorems 1 and 2 of [5], due to the first author. (shrink)

We determine the large cardinal consistency strength of the existence of a λ-supercompact cardinal κ such that equation image fails at λ. Indeed, we show that the existence of a λ-supercompact cardinal κ such that 2λ ≥ θ is equiconsistent with the existence of a λ-supercompact cardinal that is also θ-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.

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.

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}-supercompact, for any desired θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals (...) or with the class of nearly θκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta_\kappa}$$\end{document}-supercompact cardinals κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document}, for nearly any desired function κ↦θκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa\mapsto\theta_\kappa}$$\end{document}. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals. (shrink)

In this paper, we prove that: if κ is supercompact and the math formula Hypothesis holds, then there is a proper class of regular cardinals in math formula which are measurable in math formula. Woodin also proved this result independently [11]. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the math formula Hypothesis and supercompact cardinals, large cardinals in math formula are reflected to be large cardinals in math formula (...) in a local way, and reveals the huge difference between math formula-supercompact cardinals and supercompact cardinals under the math formula Hypothesis. (shrink)

We characterize some large cardinal properties, such as μ-measurability and P 2 (κ)-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on P κ (2 κ ). This leads to the characterization of 2 κ -supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, Full κ , of P κ (2 κ ), whose elements code measures on cardinals less than (...) κ. The hypothesis that Full κ is stationary (a weaker assumption than 2 κ -supercompactness) is equivalent to a higher order Lowenheim-Skolem property, and settles a question about directed versus chain-type unions on P κ λ. (shrink)

Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as (...) well. The large cardinal hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either equation image or equation image. In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ-directed closed and -distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ < κ is δ+ supercompact. (shrink)

We 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.

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.

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. (shrink)