# Blog Archives

# Topic Archive: superstrong cardinals

# Normal Measures and Tall Cardinals

I will discuss the number of normal measures a non-$(\kappa + 2)$-strong tall cardinal $\kappa$ can carry, paying particular attention to the cases where $\kappa$ is either the least measurable cardinal or the least measurable limit of strong cardinals. This is joint work with James Cummings.

# 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.