Connecting Constructive Notions of Ordinals in Homotopy Type Theory

Authors Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu

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Author Details

Nicolai Kraus
  • University of Nottingham, UK
Fredrik Nordvall Forsberg
  • University of Strathclyde, UK
Chuangjie Xu
  • fortiss GmbH, München, Germany


We thank the participants of the conferences Developments in Computer Science and TYPES, as well as Helmut Schwichtenberg and Thorsten Altenkirch for fruitful discussions on this work. We are also grateful to the anonymous reviewers, whose remarks helped us improve the paper.

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Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu. Connecting Constructive Notions of Ordinals in Homotopy Type Theory. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Constructive ordinals
  • Cantor normal forms
  • Brouwer trees


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