Computability on the space of countable ordinals

TL;DR

This paper develops a notion of computability on the space of countable ordinals using a representation framework, enabling effective analysis of classical theorems and operators within this context.

Contribution

It introduces a formal definition of computability on countable ordinals and applies it to effective descriptive set theory and the Weihrauch lattice.

Findings

Characterization of computability via four key operations

Computable versions of Lusin separation and Hausdorff-Kuratowski theorems

Definition of an ordinal iteration operator in the Weihrauch lattice

Abstract

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of countable ordinals via a representation in the sense of computable analysis. The computability structure is characterized by the computability of four specific operations, and we prove further relevant operations to be computable. Some alternative approaches are discussed, too. As an application in effective descriptive set theory, we can then state and prove computable uniform versions of the Lusin separation theorem and the Hausdorff-Kuratowski theorem. Furthermore, we introduce an operator on the Weihrauch lattice corresponding to iteration of some principle over a countable ordinal.