Connecting Constructive Notions of Ordinals in Homotopy Type Theory

TL;DR

This paper explores three different constructive notions of ordinals in homotopy type theory, analyzing their properties, relationships, and formalizations in cubical Agda, highlighting their decidability and structural similarities.

Contribution

It introduces and compares three notions of ordinals in homotopy type theory, establishing their relationships and formalizing the results in cubical Agda.

Findings

All three ordinal notions are extensional and wellfounded.

Most properties are decidable for Cantor normal forms.

Embeddings preserve structure between the notions.

Abstract

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.