James sequences and Dependent Choices | Zendy

Morillon Marianne | Zendy

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James sequences and Dependent Choices

Author(s) -

Morillon Marianne

Publication year - 2005

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.200410017

Subject(s) - mathematics , axiom of choice , dedekind cut , compact space , discrete mathematics , axiom , ideal (ethics) , set theory , continuum hypothesis , combinatorics , pure mathematics , set (abstract data type) , mathematical analysis , epistemology , computer science , philosophy , geometry , programming language

We prove James's sequential characterization of (compact) reflexivity in set‐theory ZF + DC , where DC is the axiom of Dependent Choices. In turn, James's criterion implies that every infinite set is Dedekind‐infinite, whence it is not provable in ZF . Our proof in ZF + DC of James' criterion leads us to various notions of reflexivity which are equivalent in ZFC but are not equivalent in ZF . We also show that the weak compactness of the closed unit ball of a (simply) reflexive space does not imply the Boolean Prime Ideal theorem : this solves a question raised in [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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