Small Forcing Makes Any Cardinal Superdestructible

TL;DR

This paper demonstrates that small forcing destroys the indestructibility of supercompact cardinals, making any cardinal superdestructible, yet indestructible cardinals can still be resurrected but not strongly so.

Contribution

It establishes that small forcing universally induces superdestructibility in any cardinal and clarifies the limits of resurrection for indestructible cardinals.

Findings

Small forcing destroys indestructibility of supercompact cardinals.

Any cardinal becomes superdestructible after small forcing.

Indestructible cardinals remain resurrectible but not strongly resurrectible.

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.