Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q} \in V_\theta}$$\end{document}, the cardinal κ will exhibit none of the large cardinal properties with target θ or larger. (shrink)

The C (n)-cardinals were introduced recently by Bagaria and are strong forms of the usual large cardinals. For a wide range of large cardinal notions, Bagaria has shown that the consistency of the corresponding C (n)-versions follows from the existence of rank-into-rank elementary embeddings. In this article, we further study the C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardinals, giving some improved consistency bounds while, at the same time, addressing questions which had been left open. In addition, we (...) consider two cases which were not dealt with by Bagaria; namely, C (n)-Woodin and C (n)-strongly compact cardinals, for which we provide characterizations in terms of their ordinary counterparts. Finally, we give a brief account on the interaction of C (n)-cardinals with the forcing machinery. (shrink)

The resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone, who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin’s Maximum++. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, (...) for example, failures of the weak square principle. (shrink)

We give a characterization of extendibility in terms of embeddings between the structures H λ . By that means, we show that the GCH can be forced (by a class forcing) while preserving extendible cardinals. As a corollary, we argue that such cardinals cannot in general be made indestructible by (set) forcing, under a wide variety of forcing notions.

In this note, we start with the notion of a superhuge cardinal and strengthen it by requiring that the elementary embeddings witnessing this property are, in addition, sufficiently superstrong above their target. This modification leads to a new large cardinal which we call ultrahuge. Subsequently, we study the placement of ultrahugeness in the usual large cardinal hierarchy, while at the same time show that some standard techniques apply nicely in the context of ultrahuge cardinals as well.

Weak filters were introduced by K. Schlechta in the ’90s with the aim of interpreting defaults via a generalized ‘most’ quantifier in first-order logic. They arguably represent the largest class of structures that qualify as a ‘collection of large subsets’ of a given index set |$I$|⁠, in the sense that it is difficult to think of a weaker, but still plausible, definition of the concept. The notion of weak ultrafilter naturally emerges and has been used in epistemic logic and other (...) knowledge representation (KR) applications. We provide a comprehensive exposition of weak filters and ultrafilters, comparing them with their classical counterparts that have found very important applications in logic, set theory and topology. Weak (ultra)filters capture the ‘majorities’ of social choice theory (which coincide with the commonsense understanding of a ‘large subset’) and in that respect, they outperform classical (ultra)filters in their role as ‘collections of large subsets’. Yet, they lack some of the elegant properties of classical (ultra)filters that make them an almost perfect match for logical theories. We investigate the extent to which some important classical results carry through in this new setting, and we focus on genuinely weak filters and genuinely weak ultrafilters. For weak ultrafilters, we proceed to provide concrete examples, answer questions of existence and characterize their construction. The class of weak (ultra)filters represents a genuine contribution of KR to set theory that might be of some interest to set theorists too and we initiate this study by providing a glimpse on natural set-theoretic questions. (shrink)