{"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/church-s-thesis-and-related-axioms-in-coq-s","title":"Church's thesis and related axioms in Coq's type theory","arxiv_id":"2009.00416","date":"2020-09-01","proceeding":null,"authors":["Yannick Forster"],"abstract":"\"Church's thesis\" ($\\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\\mathbb{N} \\to \\mathbb{N}$ is computable, i.e. definable in a model of computation. $\\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak K\\\"onig's Lemma and the fan theorem, respectively. Recently, $\\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak K\\\"onig's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that $\\mathsf{CT}$ is inconsistent only with a combination of classical logic and choice axioms. We study consequences of $\\mathsf{CT}$ and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.","url_abs":"https://arxiv.org/abs/2009.00416v1","url_pdf":"https://arxiv.org/pdf/2009.00416v1.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":"church-s-thesis-and-related-axioms-in-coq-s","repo_url":"https://github.com/uds-psl/churchs-thesis-coq","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}