Blass, Andreas 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 Cited in 2 ReviewsCited in 8 Documents MSC: 03E25 Axiom of choice and related propositions 15A03 Vector spaces, linear dependence, rank, lineability Keywords:basis of a vector space; ZF without regularity; urelements; axiom of multiple choice; AC Citations:Zbl 0544.00006; Zbl 0148.254 PDF BibTeX XML OpenURL