The noneffectivity of Arslanov's completeness criterion and related theorems

TL;DR

This paper investigates the limitations of Arslanov's completeness criterion and related theorems, demonstrating the failure of certain parameterized versions and analyzing their implications in computability theory.

Contribution

It provides a detailed analysis showing the non-effectivity of Arslanov's criterion and explores the failure of parameterized extensions of the recursion theorem.

Findings

Parameterized version of Arslanov's criterion fails

Parameterized version of Visser's ADN theorem fails

Highlights limitations in computability theory extensions

Abstract

We discuss the (non)effectivity of Arslanov's completeness criterion. In particular, we show that a parameterized version, similar to the recursion theorem with parameters, fails. We also discuss the parameterized version of another extension of the recursion theorem, namely Visser's ADN theorem.