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The cofinality of the least Berkeley cardinal and the extent of dependent choice
Author(s) -
Cutolo Raffaella
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800006
Subject(s) - cofinality , axiom , mathematics , mathematical economics , regular cardinal , discrete mathematics , uncountable set , geometry , countable set
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 = ω 1 . In particular, it will follow that the cofinality of the least Berkeley is independent of ZF .