Game arguments in some existence theorems of Friedberg numberings

TL;DR

This paper introduces game-theoretic proofs for key theorems on Friedberg numberings, providing new insights into their structure and relationships within computability theory.

Contribution

It offers novel game-theoretic proofs for the existence and properties of Friedberg numberings, including incomparable and independent sequences.

Findings

Proves existence of two incomparable Friedberg numberings

Establishes a uniformly c.e. sequence of pairwise incomparable Friedberg numberings

Demonstrates a uniformly c.e. independent sequence of Friedberg numberings

Abstract

We provide game-theoretic proofs of some well-known existence theorems of Friedberg numberings for the class of all partial computable functions, including (1) the existence of two incomparable Friedberg numberings; (2) the existence of a uniformly c.e. sequence of pairwise incomparable Friedberg numberings; (3) the existence of a uniformly c.e. independent sequence of Friedberg numberings. Parameterizing these proofs, we have game-theoretic proofs of Kummer's criteria and their modifications.