Set-theoretic and type-theoretic ordinals coincide

de Jong, Tom and Kraus, Nicolai and Nordvall Forsberg, Fredrik and Xu, Chuangjie (2023) Set-theoretic and type-theoretic ordinals coincide. In: Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science, 2023-06-26 - 2023-06-29.

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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 typetheoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.

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