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

Blass, Andreas

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

Keywords:

basis of a vector space; ZF without regularity; urelements; axiom of multiple choice; AC

Citations:

Zbl 0544.00006; Zbl 0148.254