{"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/set-theoretic-and-type-theoretic-ordinals","title":"Set-Theoretic and Type-Theoretic Ordinals Coincide","arxiv_id":"2301.10696","date":"2023-01-25","proceeding":null,"authors":["Tom de Jong","Nicolai Kraus","Fredrik Nordvall Forsberg","Chuangjie Xu"],"abstract":"In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.","url_abs":"https://arxiv.org/abs/2301.10696v3","url_pdf":"https://arxiv.org/pdf/2301.10696v3.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":"set-theoretic-and-type-theoretic-ordinals","repo_url":"https://github.com/martinescardo/TypeTopology","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"set-theoretic-and-type-theoretic-ordinals","repo_url":"https://github.com/tnttodda/typetopology","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}