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 $\operatorname {\mathrm {cf}}(\nu )$ 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. (shrink)

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

We establish the consistency of the failure of the diamond principle on a cardinal [Formula: see text] which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of [Formula: see text] in a Radin generic extension.

We study the Magidor iteration of Prikry forcings, and the resulting normal measures on 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}, the first measurable cardinal in a generic extension. We show that when applying the iteration to a core model below 0¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0^{\P}}$$\end{document}, then there exists a natural correspondence between the normal measures on 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} in the ground model, and those (...) of the generic extension. (shrink)

We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.

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.

Letκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs = λ starting from a ground model in whicho = λ and prove that assuming ¬0¶,s = λ implies thato ≥ λ in the core model.

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on [Formula: see text], if [Formula: see text] is weakly compact, then [Formula: see text] holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails (...) at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [O. B. -Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19(1) (2019) 1950002]. (shrink)

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our (...) main results are in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and there is no such sequence of length $$\omega _1$$ ω 1. (shrink)

We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak {s}_\theta, \mathfrak {p}_\theta, \mathfrak {t}_\theta, \mathfrak {g}_\theta, \mathfrak {r}_\theta $ at uncountable regular cardinals $\theta $. Motivated by a theorem of Raghavan–Shelah who proved that $\mathfrak {s}_\theta \leq \mathfrak {b}_\theta $, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak {s}_\theta $ with $\mathfrak {p}_\theta $ and $\mathfrak {g}_\theta $. In the second part of the paper we study variations (...) of the extender-based Radin forcing to establish several consistency results concerning $\mathfrak {r}_\theta,\mathfrak {s}_\theta $ from hyper-measurability assumptions, results which were previously known to be consistent only from supercompactness assumptions. In doing so, we answer questions from [1], [15] and [7], and improve the large cardinal strength assumptions for results from [10] and [3]. (shrink)