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# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## HOD-supercompactness, Indestructibility, and Level by Level Equivalence

### Tom 62 / 2014

Bulletin Polish Acad. Sci. Math. 62 (2014), 197-209 MSC: 03E35, 03E55. DOI: 10.4064/ba62-3-1

#### Streszczenie

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 $\kappa _0$, $\kappa _0$ 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 $\kappa _0$ but fails below $\kappa _0$. 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.

#### Autorzy

• Arthur W. ApterDepartment of Mathematics
Baruch College of CUNY
New York, NY 10010, U.S.A.
and
365 Fifth Avenue
New York, NY 10016, U.S.A.
e-mail
• Shoshana FriedmanDepartment of Mathematics and Computer Science
Kingsborough Community College-CUNY
2001 Oriental Blvd, Brooklyn, NY 11235, U.S.A.
e-mail

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