TY - JOUR

T1 - Decidability and undecidability of theories with a predicate for the primes.

AU - Bateman, P.T.

AU - Jockusch, C.G

AU - Woods, Alan

PY - 1993

Y1 - 1993

N2 - It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure , 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 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.

AB - It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure , 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 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.

U2 - 10.2307/2275227

DO - 10.2307/2275227

M3 - Article

VL - 58

SP - 672

EP - 687

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

ER -