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In an attempt to extend the property of being supercompact but not hod-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not hod-supercompact holds for the least supercompact cardinal κ0, κ0 is indestructibly supercompact, the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ0 but fails below κ0. Additionally, we get the property of being supercompact but not hod-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness holds.

HOD-supercompactness, Indestructibility, and Level by Level Equivalence