INDESTRUCTIBILITY WHEN THE FIRST TWO MEASURABLE CARDINALS ARE STRONGLY COMPACT

Abstract

Abstract We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$ , we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible (i.e., $\kappa _1$ ’s strong compactness is indestructible under arbitrary $\kappa _1$ -directed closed forcing), and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $ . This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.