From: John Rickard Subject: Re: Proof of Result in R(n) Date: 14 Aug 2000 19:03:15 +0100 (BST) Newsgroups: sci.math Summary: A basis for every vector space, and the Axiom of Choice Richard Carr wrote: : Where does "every vector space has a basis" fit in? I don't see it in : "Equivalents Of The Axiom Of Choice" (unless it is disguised in the : algebra part). Is it weaker than AC? According to P. T. Johnstone ("Notes on logic and set theory", Cambridge University Press 1987, ISBN 0-521-33692-9), A. R. Blass proved in 1983 that AC is equivalent to "every vector space has a basis", but it is (was?) not known whether, for example, AC is equivalent to "every vector space over R has a basis". (Johnstone's book also says that J. D. Halpern proved in 1966 that, for every field F, AC is equivalent to "every linearly independent set in a vector space over F can be extended to a basis"; the proof is sketched in an exercise in Johnstone's book.) -- John Rickard