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arXiv · 2609.20140

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

Gabriel Fernandes·Renan Maneli Mezabarba·Vinicius de Oliveira Rodrigues
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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.

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Gabriel Fernandes, Renan Maneli Mezabarba, Vinicius de Oliveira Rodrigues. 2026-09-17. Existence of bases implies the axiom of choice, a foundation-free proof. https://arxiv.org/abs/2609.20140

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