Existence of bases implies the axiom of choice, a foundation-free proof

Abstract

We prove that, in Zermelo--Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice, concluding that the classical equivalence between $\AC$ and the existence of bases does not require the Axiom of Foundation. More specifically, we prove that if every vector space over a field of characteristic zero has a basis, then $\AC$ holds. This result extends to set theory with atoms.