James sequences and Dependent Choices

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].