We show the relative consistency of the existence of two strongly compact cardinals κ₁ and κ₂ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ₁. In the model constructed, κ₁'s strong compactness is indestructible under arbitrary κ₁-directed closed forcing, κ₁ is a limit of measurable cardinals, κ₂'s strong compactness is indestructible under κ₂-directed closed forcing which is also (κ₂,∞)-distributive, and κ₂ is fully supercompact.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-2-2,
author = {Arthur W. Apter},
title = {Indestructibility, strong compactness, and level by level equivalence},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {113-126},
zbl = {1186.03067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-2-2}
}
Arthur W. Apter. Indestructibility, strong compactness, and level by level equivalence. Fundamenta Mathematicae, Tome 205 (2009) pp. 113-126. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-2-2/