Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ exhibits mixed levels of indestructibility. Specifically, κ 's strong compactness, but not its supercompactness, is indestructible under any κ -directed closed forcing which also adds a Cohen subset of κ. On the other hand, in this model, κ 's supercompactness is indestructible under any κ -directed closed forcing which does not add a Cohen subset of κ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-2,
author = {Arthur W. Apter},
title = {Mixed Levels of Indestructibility},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {63},
year = {2015},
pages = {113-122},
zbl = {06526969},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-2}
}
Arthur W. Apter. Mixed Levels of Indestructibility. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) pp. 113-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-2/