Recursive functions and existentially closed structures

TL;DR

This paper explores the relationship between recursive functions, model-theoretic properties, and undecidability, demonstrating a theory where all partial recursive functions are representable without interpreting Robinson's theory R.

Contribution

It introduces a theory where all partial recursive functions are representable yet does not interpret Robinson's theory R, using model-theoretic tools to analyze properties of model completions.

Findings

Existence of a theory with all partial recursive functions representable

Such a theory does not interpret Robinson's theory R

Characterization of $orall eg$ theories interpretable in existential theories

Abstract

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$ does not interpret Robinson's theory $R$. To this end, we borrow tools from model theory--specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of $\exists\forall$ theories interpretable in existential theories in the process.