Identity crises and strong compactness

Abstract

Abstract Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n , the first n strongly compact cardinals k 1 ..… k n are so that k i ; for i = 1..… n is both the i th measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.