Decidability and classification of the theory of integers with primes

by Kaplan and Shelah. [KpSh:1082]
J Symbolic Logic, 2017
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.

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