Indestructible strong compactness and level by level inequivalence

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Abstract

If inline image are such that δ is indestructibly supercompact and γ is measurable, then it must be the case that level by level inequivalence between strong compactness and supercompactness fails. We prove a theorem which points to this result being best possible. Specifically, we show that relative to the existence of cardinals inline image such that κ1 is λ-supercompact and λ is inaccessible, there is a model for level by level inequivalence between strong compactness and supercompactness containing a supercompact cardinal inline image in which κ’s strong compactness, but not supercompactness, is indestructible under κ-directed closed forcing. In this model, κ is the least strongly compact cardinal, and no cardinal is supercompact up to an inaccessible cardinal.

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