Decidability and classification of the theory of integers with primes

TL;DR

Under Dickson's conjecture, the paper proves that the theory of integers with primes is decidable, unstable, and supersimple, contrasting with the known undecidability of a related theory.

Contribution

It establishes the decidability and model-theoretic properties of the prime-integers theory under a major conjecture, extending understanding of prime-related structures.

Findings

Th(Z,+,1,0,Pr) is decidable under Dickson's conjecture

Th(Z,+,1,0,Pr) is unstable and supersimple

Th(Z,+,0,Pr,<) is undecidable as previously known

Abstract

We show that under Dickson's conjecture about the distribution of primes in the natural numbers, the theory Th(Z,+,1,0,Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable and supersimple. This is in contrast with Th(Z,+,0,Pr,<) which is known to be undecidable by the works of Jockusch, Bateman and Woods.