Type-Theoretic Approaches to Ordinals
release_nfut227d35atjgn7r565y4ejzu
by
Nicolai Kraus and Fredrik Nordvall Forsberg and Chuangjie Xu
2022
Abstract
In a constructive setting, no concrete formulation of ordinal numbers can
simultaneously have all the properties one might be interested in; for example,
being able to calculate limits of sequences is constructively incompatible with
deciding extensional equality. Using homotopy type theory as the foundational
setting, we develop an abstract framework for ordinal theory and establish a
collection of desirable properties and constructions. We then study and compare
three concrete implementations of ordinals in homotopy type theory: first, a
notation system based on Cantor normal forms (binary trees); second, a refined
version of Brouwer trees (infinitely-branching trees); and third, extensional
well-founded orders.
Each of our three formulations has the central properties expected of
ordinals, such as being equipped with an extensional and well-founded ordering
as well as allowing basic arithmetic operations, but they differ with respect
to what they make possible in addition. For example, for finite collections of
ordinals, Cantor normal forms have decidable properties, but suprema of
infinite collections cannot be computed. In contrast, extensional well-founded
orders work well with infinite collections, but almost all properties are
undecidable. Brouwer trees take the sweet spot in the middle by combining a
restricted form of decidability with the ability to work with infinite
increasing sequences.
Our three approaches are connected by canonical order-preserving functions
from the "more decidable" to the "less decidable" notions. We have formalised
the results on Cantor normal forms and Brouwer trees in cubical Agda, while
extensional well-founded orders have been studied and formalised thoroughly by
Escardo and his collaborators. Finally, we compare the computational efficiency
of our implementations with the results reported by Berger.
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