Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Arthur W. Apter¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 3, pp. 189-197

Résumé

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”.

Détail

DOI : 10.4064/ba57-3-1

Keywords: provide upper lower bounds consistency strength theories neg omega successor cardinals except successors uncountable limit cardinals regular every uncountable limit cardinal singular successor every uncountable limit cardinal singular cofinality omega negac omega successor cardinals except successors uncountable limit cardinals regular every uncountable limit cardinal singular successor every uncountable limit cardinal singular cofinality omega particular models these theories satisfy negac omega kappa singular kappa either uncountable limit cardinal successor uncountable limit cardinal