Institute of Mathematics / Publishing house / Journals and Serials / Bulletin of the Polish Academy of Sciences. Mathematics / All issues

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”.