# 2007 seminar talk: Indestructible cardinals and forcing axioms

Talk held by Thomas Johnstone (City University of New York) at the KGRC seminar on 2007-05-10.

### Abstract

Determining which cardinals can be made indestructible by which classes of forcing has been a major interest in modern set theory. Inspired by Laver's celebrated result for supercompact cardinals, I will present a method of making strongly unfoldable cardinals indestructible. These cardinals strengthen weakly compact and indescribable cardinals, yet they are rather small in the hierarchy of large cardinals, as they are consistent with V=L. Starting with a strongly unfoldable cardinal kappa, I will produce a forcing extension V[G], in which the strong unfoldability of kappa is indestructible by all <kappa-closed, kappa+ preserving posets. In particular, the weak compactness and indescribability of kappa is indestructible. Previously known results would have had to assume the existence of a strong or supercompact cardinal to obtain this general indestructibility. Combining the method with the idea of Baumgartner's proof of the relative consistency of the Proper Forcing Axiom PFA, I will establish the consistency of a weakening of PFA relative to the existence of a strongly unfoldable cardinal. I will also discuss several related open questions. Part of the material in this talk is joint work with Joel David Hamkins.