HOD-supercompactness, Indestructibility, and Level by Level Equivalence
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.
- Department of Mathematics, Baruch College of CUNY, New York, NY 10010, U.S.A.
- The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, NY 10016, U.S.A.
- Department of Mathematics and Computer Science, Kingsborough Community College-CUNY, 2001 Oriental Blvd, Brooklyn, NY 11235, U.S.A.