Recursive functions and existentially closed structures

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 [Formula: see text] in which all partially recursive functions are representable, yet [Formula: see text] does not interpret Robinson’s theory [Formula: see text]. 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 [Formula: see text] theories interpretable in existential theories in the process.