Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Bulletin of the Polish Academy of Sciences. Mathematics / All issues

Abstract

Starting from a supercompact cardinal $\kappa$, we force and construct a model in which $\kappa$ is both the least strongly compact and least supercompact cardinal and $\kappa$ exhibits mixed levels of indestructibility. Specifically, $\kappa$'s strong compactness, but not its supercompactness, is indestructible under any $\kappa$-directed closed forcing which also adds a Cohen subset of $\kappa$. On the other hand, in this model, $\kappa$'s supercompactness is indestructible under any $\kappa$-directed closed forcing which does not add a Cohen subset of $\kappa$.