In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to ‮א‬₁. Later we give applications, among them the consistency of MM with ‮א‬ω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.

We construct a model with a saturated ideal _I_ over \({\mathcal {P}}_{\kappa }\lambda \) and study the extent of saturation of _I_.

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...) order ω need not imply Refl1. Moreover, we prove that if κ is -weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set W⊆κ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down. (shrink)

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...) It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

In this paper we investigate the consistency and consequences of the downward Löwenheim–Skolem–Tarski theorem for extension of the first order logic by the Magidor–Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang’s conjecture at successor of singular 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 consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

The stable core, an inner model of the form $\langle L[S],\in, S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete (...) proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in V, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [29]. (shrink)

We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

We develop the theory of layered posets and use the notion of layering to prove a new iteration theorem is κ-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications:1 For any n≥1, a wide variety of <ωn−1-closed, ωn+1-cc posets of size ωn+1 can consistently (...) be absorbed by quotients of saturated ideals on ωn.2 For any n∈ω, the tree property at ωn+3 is consistent with Chang’s conjecture ↠. (shrink)

We introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2.

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ -strategically closed forcing and λ is weakly compact, then we show thatA = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets}must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a (...) Mahlo cardinal in which the least supercompact cardinal κ is also the least strongly compact cardinal, κ 's strongness is indestructible under κ -strategically closed forcing, κ 's supercompactness is indestructible under κ -directed closed forcing not adding any new subsets of κ, and δ is Mahlo and reflects stationary sets iff δ is weakly compact. In this model, no strong cardinal δ < κ is indestructible under δ -strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal κ whose strongness is indestructible under κ -strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under κ -directed closed forcing not adding any new subsets of κ. (shrink)

We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of ZFC \ CH. In this paper we present a sufficient condition to be consistently the set of cardinalities of branches of Kurepa trees.

For an ordinal \, I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \-subcomplete forcings are \-subcomplete, and if \ and \ are forcing-equivalent notions, then \ is \-subcomplete iff \ is. I formulate a Two Step Theorem for \-subcompleteness and prove an RCS iteration theorem for \-subcompleteness (...) which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \-subcompleteness follows from the \-distributivity of a forcing, for \. (shrink)

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a stationary (...) subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

Consider where κ is an uncountable regular cardinal. By a result of Shelah's we have for almost all witnessing this. Here we consider the question if there could be a similar result for. We will discuss some basic facts implying that this cannot hold in general. We will use these facts to construct an interesting example of a pseudo Prikry forcing, answering a question of Sinapova.

We study the structural rainbow Ramsey theory at uncountable cardinals. Compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than ω1, we uncover some connections between versions of Chang's Conjectures and instances of rainbow Ramsey partition relations, addressing a question raised in [18].

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong (...) coloring principle U(. . .), introduced in Part I of this series. (shrink)

We study various aspects of the size, including the cardinality, of closed unbounded subsets of [λ]<κ, especially when λ = κ+n for n ε ω. The problem is resolved into the study of the size of certain stationary sets. Relative to the existence of an ω1-Erdös cardinal it is shown consistent that ωω3 < ωω13 and every closed unbounded subsetof [ω3]<ω2 has cardinality ωω13. A weakening of the ω1-Erdös property, ω1-remarkability, is defined and shown to be retained under a large (...) class of Easton-like forcings applied to ω1-Erdös cardinals. A class of reverse-Easton forcings preserving α-Erdösness is also described, with special attention to the establishment of □-principles. Article Outline. (shrink)

We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses.

Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ=κ, using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter F. This forcing adds a generic of minimal degree if and only if F is κ-saturated.

The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis of (...) a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > א1, then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. (shrink)

We prove, assuming the existence of large cardinals, the relative consistency of the existence of strongly saturated ideals on small cardinals. We also give some information about the problem, how many σ-additive, 0-1-valued measures over a small cardinal are necessary, such that every subset of the cardinal is measurable in at least one of them.

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa $, updating some classical constructions in the process. This includes models of $\mathsf {CSR}(\kappa ^{++})\wedge {\sf TP}(\kappa ^{++})$ (both with and without ${\sf AP}(\kappa ^{++})$ ) and models of the conjunctions ${\sf SR}(\kappa ^{++}) \wedge \mathsf {wTP}(\kappa ^{++}) \wedge {\sf AP}(\kappa ^{++})$ and $\neg {\sf AP}(\kappa ^{++}) \wedge {\sf SR}(\kappa ^{++})$ (the latter was originally obtained in joint work by Krueger and (...) the first author [9], and is here given using different methods). Analogs of these results with the failure of $\sf {SH}(\kappa ^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^\kappa $, applying recent joint work by Honzik and the third author. (shrink)