This is a talk for the CUNY Set Theory Seminar on February 1, 2013, 10:00 am.

Abstract.  Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $\kappa$ can be made indestructible by $<\kappa$-directed closed forcing and continued with the Gitik-Shelah treatment of strong cardinals, is by now nearly pervasive in set theory, nevertheless I shall show that no superstrong strong cardinal—and hence also no $1$-extendible cardinal, no almost huge cardinal and no rank-into-rank cardinal—can be made indestructible, even by comparatively mild forcing: all such cardinals $\kappa$ are destroyed by $\text{Add}(\kappa ,1)$, by $\text{Add}(\kappa ,{\kappa }^{+})$, by $\text{Add}({\kappa }^{+},1)$ and by many other commonly considered forcing notions.

This is very recent joint work with Konstantinos Tsaprounis and Joan Bagaria.

nylogic.org | Set Theory Seminar |