Set-Theoretic and Type-Theoretic Ordinals Coincide

KRAUS, NICOLAI; de Jong, Tom; Nordvall Forsberg, Fredrik; Xu, Chuangjie

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.

• Presentation Conference Type: Conference Paper (unpublished)
• Conference Name: Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• Conference Location: Boston, USA
• Start Date: Jun 26, 2023
• End Date: Jun 29, 2023
• Deposit Date: Jul 4, 2023
• Publicly Available Date: Jul 14, 2023
• Public URL: https://nottingham-repository.worktribe.com/output/22715787