Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns
Volume 57 / 2009
Abstract
We provide upper and lower bounds in consistency strength for the theories “ZF + $\neg $AC$_\omega $ + 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 $\omega $” and “ZF + $\neg$AC$_\omega $ + 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 $\omega _1$”. In particular, our models for both of these theories satisfy “ZF + $\neg$AC$_\omega $ + $\kappa $ is singular if{f} $\kappa $ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”.
