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.

This paper reconstructs Steel’s multiverse project in his ‘Gödel’s program’ (Steel [2014]), first by comparing it to those of Hamkins [2012] and Woodin [2011], then by detailed analysis what’s presented in Steel’s brief text. In particular, we reconstruct his notion of a ‘natural’ theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and isolate what it would take (...) to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel’s story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of ‘meaning’ from the account). The relevant mathematics is laid out in the appendices. (shrink)

Models of set theory are constructed where the non-stationary ideal on PΩ1λ is presaturated. The initial model has a Woodin cardinal. Using the Lévy collapse the Woodin cardinal becomes λ+ in the final model. These models provide new information about the consistency strength of a presaturated ideal onPΩ1λ for λ greater than Ω1.

The cut and choose game is one of the infinitary games on a complete Boolean algebra B introduced by Jech. We prove that existence of a winning strategy for II in implies semiproperness of B. If the existence of a supercompact cardinal is consistent then so is “for every 1-distributive algebra B II has a winning strategy in ”.

By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in (...) addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model. (shrink)

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection equation image containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in equation image. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot (...) fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω 1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary (...) ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal. (shrink)

We study Σ1-definable sets in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1-definable, the set of all stationary subsets of ω1 is not Σ1-definable and the complement of every Σ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1-definable well-ordering (...) of H and the existence of a Δ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1-definable subsets of κκ, in the case where κ itself has certain large cardinal properties. (shrink)

We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: consistently, there is a nowhere precipitous normal ideal I on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_2}$$\end{document} such that player nonempty wins the pressing down game of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} on I even if player empty starts.

Starting with a measurable cardinal κ of the Mitchell order κ++ we construct a model with a precipitous ideal on ℵ1 but without normal precipitous ideals. This answers a question by T. Jech and K. Prikry. In the constructed model there are no Q-point precipitous filters on ℵ1, i. e. those isomorphic to extensions of Cubℵ1.

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.