Set-Theoretic and Type-Theoretic Ordinals Coincide

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.

v2: Minor changes. To appear at LICS'23. v3: Acknowledgments updated

References in corpus (4)

• Type-Theoretic Approaches to Ordinals
• On Small Types in Univalent Foundations
• Constructive theory of ordinals
• The cumulative hierarchy in Homotopy Type Theory

Cited by in corpus (2)

• The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations
• Continuous and algebraic domains in univalent foundations