The complexity of completions in partial combinatory algebra

TL;DR

This paper investigates the complexity of completions in partial combinatory algebras, especially Kleene's first model, revealing the existence of low Turing degree completions and their relation to PA extensions.

Contribution

It demonstrates that while computable completions do not exist, low Turing degree completions do, and connects these to complete extensions of Peano Arithmetic.

Findings

No computable completions exist for Kleene's first model.

Low Turing degree completions are constructed.

Connections between completions and PA extensions are established.

Abstract

We discuss the complexity of completions of partial combinatory algebras, in particular of Kleene's first model. Various completions of this model exist in the literature, but all of them have high complexity. We show that although there do not exist computable completions, there exists completions of low Turing degree. We use this construction to relate completions of Kleene's first model to complete extensions of PA. We also discuss the complexity of pcas defined from nonstandard models of PA.