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Abstract

We prove two theorems, one concerning level by level inequivalence between strong compactness and supercompactness, and one concerning level by level equivalence between strong compactness and supercompactness. We first show that in a universe containing a supercompact cardinal but of restricted size, it is possible to control precisely the difference between the degree of strong compactness and supercompactness that any measurable cardinal exhibits. We then show that in an unrestricted size universe containing many supercompact cardinals, it is possible to have significant failures of GCH along with level by level equivalence between strong compactness and supercompactness, except possibly at inaccessible levels.