Recursive functions and existentially closed structures

摘要

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 there exists for all theories interpretable in existential theories in the process.