Small Forcing Makes Any Cardinal Superdestructible

Abstract

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $\kappa$ becomes superdestructible---any further $<\kappa$-closed forcing which adds a subset to $\kappa$ will destroy the measurability, even the weak compactness, of $\kappa$. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.

This paper appeared in 1998; I have finally now uploaded the article to the arxiv. Commentary concerning this article can be made on the author's blog at http://jdh.hamkins.org/superdestructibility/