On Singular Stationarity II (tight stationarity and extenders-based methods)

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in \cite{ForMag-MS}. We obtain two consistency results which show that it is possible for a sequence of regular cardinals $( \kappa_n )_{n < \omega}$ to have the property that for every sequence $\vec{S}$, of some fixed-cofinality stationary sets $S_n \subseteq \kappa_n$, $\vec{S}$ is tightly stationary in a generic extension. The results are obtained using variations of the short-extenders forcing method

Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.Comment: arXiv admin note: text overlap with arXiv:1510.0293

Joint diamonds and Laver diamonds

The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. While equivalent in the case of $\diamondsuit_\kappa$, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the large cardinals under consideration and prove sharp separations between joint Laver diamonds of different lengths in the case of $\theta$-supercompact cardinals.Comment: 34 pages; revised version with several improvements, including expanded Sections 3.3 and

Inner-model reflection principles

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the L\'evy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.Comment: 17 pages, revised version incorporating suggestions of the referees; a new co-author has been added. Commentary concerning this paper can be made at http://jdh.hamkins.org/inner-model-reflection-principle

The tree property at all regular even cardinals

Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at $\delta^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].Comment: Comments are welcom

The structure of the Mitchell order - II

We address the question regarding the structure of the Mitchell order on normal measures. We show that every well founded order can be realized as the Mitchell order on a measurable cardinal $\kappa$ from some large cardinal assumption

The maximality of the core model

If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K computes successors of weakly compact cardinals correctly. o K^c is an iterate of K. o (with Mitchell) If alpha is a cardinal > aleph_1, then K-restriction-alpha is universal for mice of height alpha. Other results in this paper, when combined with work of Woodin, imply: o If square-kappa-finite fails and kappa is a singular, strong limit cardinal, then Inductive Determinacy holds. o If square-kappa-finite fails and kappa is a weakly compact cardinal, then L(R)-determinacy holds

Ultrafilters on singular cardinals of uncountable cofinality

We prove that consistently there is a singular cardinal $\kappa$ of uncountable cofinality such that $2^\kappa$ is weakly inaccessible, and every regular cardinal strictly between $\kappa$ and $2^\kappa$ is the character of some uniform ultrafilter on $\kappa$

Stationary Reflection and the failure of SCH

In 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 $\mathrm{cf}(\nu)$ stationary subsets of $\nu^+$ reflects simultaneously. For uncountable cofinality, 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 $\mathrm{cf}(\nu) = \omega$ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.Comment: 23 page

Laver and set theory

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