Constructive Ordinal Exponentiation

TL;DR

This paper investigates the limitations of constructive ordinal exponentiation, proposing two near-constructive definitions that preserve key algebraic properties and are mechanized in Agda within homotopy type theory.

Contribution

It demonstrates the impossibility of general constructive ordinal exponentiation and introduces two alternative definitions with proven algebraic properties.

Findings

Two definitions are equivalent when applicable

Algebraic laws and cancellation properties hold for the definitions

The constructions preserve decidability of the exponential

Abstract

Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core principles of modern programming language theory such as recursion. In classical mathematics, ordinal arithmetic is well-studied; constructively, where ordinals are taken to be transitive, extensional, and wellfounded orders on sets, addition and multiplication are well-known. We present a negative result showing that general constructive ordinal exponentiation is impossible, but we suggest two definitions that come close. The first definition is abstract and solely motivated by the expected equations; this works as long as the base of the exponential is positive. The second definition is based on decreasing lists and can be seen as a constructive version of Sierpi\'nski's definition via functions with finite support; this requires the base to have a trichotomous least element. Whenever it makes sense to ask the question, the two constructions are equivalent, allowing us to prove algebraic laws, cancellation properties, and preservation of decidability of the exponential. The core ideas do not depend on any specific constructive set theory or type theory, but a concrete computer-checked mechanization using the Agda proof assistant is given in homotopy type theory.