On λ-Definable Functions on Ordinals

Abstract

We are going to generalize classical λ-calculus to the ordinal domain. Our reasoning is centered around a generalization of Church numerals, i.e., terms that define the n-fold application of their first argument to the second, to numerals for transfinite ordinals. Once the new class of ordinal λ-terms is established, we define a transfinite procedure to assign to a given ordinal λ-term a normal form if one exists. This normal form procedure is compatible with the classical case, i.e., will find normal forms for classical terms whenever they exist. We go on to prove a confluence property for our procedure. The calculus thus defined is tied into the existing framework of ordinal computability: Using our terms to define a class of functions on the ordinals, we show that this class is identical with the class of $\mathbf\Sigma_1(L)$ definable functions on Ord. This paper takes the form of an ‘extended abstract’: The technical details of the main definition, detailed examples, as well as proofs of the theorems are omitted for brevity.