More on HOD-supercompactness

Annals of Pure and Applied Logic 172 (3):102901 (2021)
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We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact.



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Suitable extender models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
Gap forcing: Generalizing the lévy-Solovay theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.
The Ground Axiom.Jonas Reitz - 2007 - Journal of Symbolic Logic 72 (4):1299 - 1317.

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