DECIDABILITY AND UNDECIDABILITY OF THEORIES WITH A PREDICATE FOR THE PRIMES

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure <omega; +, P>, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of <omega; S, P> is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semenov on decidability of monadic theories, and a proof of Semenov's result is presented.