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Existence of bases implies the axiom of choice. (English) Zbl 0557.03030

Axiomatic set theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1983, Contemp. Math. 31, 31-33 (1984).
[For the entire collection see Zbl 0544.00006.]
It is shown that in WZF (ZF without regularity and admitting urelements) the axiom of multiple choice follows from the assumption that every vector space has a basis. Hence, in full ZF, AC follows, which partially refutes a conjecture of J. D. Halpern [Proc. Am. Math. Soc. 17, 670-673 (1966; Zbl 0148.254)].
Reviewer: F.R.Drake

MSC:

03E25 Axiom of choice and related propositions
15A03 Vector spaces, linear dependence, rank, lineability