Computability in partial combinatory algebras

TL;DR

This paper explores fundamental aspects of computability within partial combinatory algebras, challenging previous assumptions and clarifying the properties of numberings and extensionality.

Contribution

It disproves Kreisel's suggestion regarding Friedberg numberings for extensional pca's and analyzes properties like separability and precompleteness.

Findings

Disproved Kreisel's suggestion about Friedberg numberings

Showed precomplete numberings are not necessarily 1-1

Discussed elements without total extensions in pca's

Abstract

We prove a number of elementary facts about computability in partial combinatory algebras (pca's). We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca's. We then discuss separability and elements without total extensions. We relate this to Ershov's notion of precompleteness, and we show that precomplete numberings are not 1-1 in general.