James sequences and Dependent Choices

Mathematical Logic Quarterly 51 (2):171-186 (2005)

We prove James's sequential characterization of 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 reflexive space does not imply the Boolean Prime Ideal theorem : this solves a question raised in [6]
Keywords weak topology  reflexive Banach space  Axiom of Choice  Dependent Choices  James' criterion  well‐founded tree
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DOI 10.1002/malq.200410017
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[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Definability of Measures and Ultrafilters.David Pincus & Robert M. Solovay - 1977 - Journal of Symbolic Logic 42 (2):179-190.

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