Indestructibility, strong compactness, and level by level equivalence

Abstract

We show the relative consistency of the existence of two strongly compact cardinals •1 and •2 which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for •1. In the model constructed, •1’s strong compactness is indestructible under arbitrary •1-directed closed forcing, •1 is a limit of measurable cardinals, •2’s strong compactness is indestructible under •2-directed closed forcing which is also (•2;1)-distributive, and •2 is fully supercompact.