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PreprintCC BY 4.0arXiv

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

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Cite

BibTeXRIS
BibTeX
@misc{fernandes2026existence,
  title = {{Existence of bases implies the axiom of choice, a foundation-free proof}},
  author = {Gabriel Fernandes and Renan Maneli Mezabarba and Vinicius de Oliveira Rodrigues},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20140},
  eprint = {2609.20140},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20140},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Existence of bases implies the axiom of choice, a foundation-free proof
AU  - Gabriel Fernandes
AU  - Renan Maneli Mezabarba
AU  - Vinicius de Oliveira Rodrigues
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.20140
UR  - https://arxiv.org/abs/2609.20140
AB  - 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 (AC), concluding that the classical equivalence between AC and the existence of bases does not require regularity. This result extends to set theory with atoms.
ER  -