Exactly controlling the non-supercompact strongly compact cardinals

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [6], due to the first author. 1 Introducing the Main Question The notions of strongly compact and supercompact cardinal are very close, so close that years ago it was an open question whether they were equivalent. When Solovay first defined the supercompact 2000 Mathematics Subject Classifications: 03E35, 03E55.

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