Universal indestructibility for degrees of supercompactness and strongly compact cardinals

We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness. In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly compact cardinals, each of which exhibits a significant amount of indestructibility for its strong compactness. The first author’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. The first author wishes to thank James Cummings for helpful discussions on the subject matter of this paper. In addition, both authors wish to thank the referee, for many helpful comments and suggestions which were incorporated into the current version of the paper.