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Sandwiching the consistency strength of two global choiceless cardinal patterns

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 3-4

189-197

We provide upper and lower bounds in consistency strength for the theories "ZF + ¬ACω + 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 + ¬ACω + 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 1". In particular, our models for both of these theories satisfy "ZF + ¬ACω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal".