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 -