# Blog Archives

# Topic Archive: extendible cardinals

# Superstrong cardinals are never Laver indestructible, and neither are extendible, almost huge and rank-into-rank cardinals

Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $kappa$ can be made indestructible by $ltkappa$-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 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 $Add(kappa,1)$, by $Add(kappa,kappa^+)$, by $Add(kappa^+,1)$ and by many other commonly considered forcing notions.

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