**April 24**

Arthur Apter,
CUNY

**Indestructibility and the First Two Strongly Compact Cardinals**

Starting from a model of ZFC with two supercompact cardinals, I will discuss how to force and construct a model in which the first two strongly compact cardinals $\kappa_1$ and $\kappa_2$ are also the first two measurable cardinals. In this model, $\kappa_1$'s strong compactness is indestructible under arbitrary $\kappa_1$-directed closed forcing, and $\kappa_2$'s strong compactness is indestructible under ${\rm Add}(\kappa_2, \lambda)$ for any ordinal $\lambda$. This answers a generalized version of a question of Sargsyan.