Connecting Constructive Notions of Ordinals in Homotopy Type Theory

Kraus, Nicolai; Nordvall Forsberg, Fredrik; Xu, Chuangjie

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.

• Conference Name: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
• Conference Location: Tallinn, Estonia
• Start Date: Aug 23, 2021
• End Date: Aug 27, 2021
• Acceptance Date: Jun 29, 2021
• Online Publication Date: Aug 23, 2021
• Publication Date: Aug 23, 2021
• Deposit Date: Jun 30, 2021
• Publicly Available Date: Aug 23, 2021
• Volume: 202
• Series Title: Leibniz International Proceedings in Informatics
• Series ISSN: 1868-8969
• Public URL: https://nottingham-repository.worktribe.com/output/5748523