The cofinality of the least Berkeley cardinal and the extent of dependent choice.

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of AC. In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐DC fails. We shall prove that this result is optimal for γ=ω or γ=ω1. In particular, it will follow that the cofinality of the least Berkeley is independent of ZF. [ABSTRACT FROM AUTHOR]