We provide upper and lower bounds in consistency strength for the theories “ZF + + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω₁”. In particular, our models for both of these theories satisfy “ZF + + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-1,
author = {Arthur W. Apter},
title = {Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {57},
year = {2009},
pages = {189-197},
zbl = {1196.03068},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-1}
}
Arthur W. Apter. Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) pp. 189-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-1/