Unconditional decidability of Presburger arithmetic with the prime predicate

Determine whether the first-order theory of \(\langle\mathbb{N};0,1,<,+,P\rangle\), where \(P\) is the set of prime numbers, is decidable without assuming Dickson’s conjecture.

Background

The paper identifies the theory of Presburger arithmetic expanded by the set of primes as its biggest open problem in the direction of applying randomness-based counter-machine simulations. The theory is known to be undecidable conditional on Dickson’s conjecture, because multiplication is then definable. An unconditional result would require converting an established theorem about primes into a proof that the structure simulates counter machines.