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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-1,
author = {Arthur W. Apter and Shoshana Friedman},
title = {HOD-supercompactness, Indestructibility, and Level by Level Equivalence},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {62},
year = {2014},
pages = {197-209},
zbl = {1311.03073},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-1}
}
Arthur W. Apter; Shoshana Friedman. HOD-supercompactness, Indestructibility, and Level by Level Equivalence. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) pp. 197-209. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba62-3-1/