Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Abstract

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 ω1”. 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”. There are many instances in the literature where choiceless large cardinal patterns are initially forced from strong hypotheses which one later sees can be weakened somewhat. For example, it is shown in [4] that the models constructed in [11] from an almost huge cardinal can actually be ∗2000 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55. †