Work

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

Year: 2026

Type: preprint

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 ($\mathsf{AC}$), concluding that the classical equivalence between $\mathsf{AC}$ and the existence of bases does not require regularity. This result extends to set theory with atoms.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.20140

Authors: Fernandes, Gabriel, Mezabarba, Renan Maneli, Rodrigues, Vinicius de Oliveira

Institution: Universidade de São Paulo

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Fundação de Amparo à Pesquisa do Estado de São Paulo

OpenAlex ID: W7213609998

Cites: 13

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Related to: 10