{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/formalization-of-the-axiom-of-choice-and-its","title":"Formalization of the Axiom of Choice and its Equivalent Theorems","arxiv_id":"1906.03930","date":"2019-06-10","proceeding":null,"authors":["Tianyu Sun","Wensheng Yu"],"abstract":"In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle, Zermelo's postulate, Zorn's lemma and the well-ordering theorem. We prove the above theorems by the axiom of choice in turn, and finally prove the axiom of choice by Zermelo's postulate and the well-ordering theorem, thus completing the cyclic proof of equivalence between them. The proofs are checked formally using the Coq proof assistant in which Morse-Kelley set theory is formalized. The whole process of formal proof demonstrates that the Coq-based machine proving of mathematics theorem is highly reliable and rigorous. The formal work of this paper is enough for most applications, especially in set theory, topology and algebra.","url_abs":"http://arxiv.org/abs/1906.03930v1","url_pdf":"http://arxiv.org/pdf/1906.03930v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"formalization-of-the-axiom-of-choice-and-its","repo_url":"https://github.com/styzystyzy/Axiom_of_Choice","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}