Journal of Symbolic Logic

Decidability and Undecidability of Theories with a Predicate for the Primes

Abstract

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure $\langle \omega; +, P\rangle$, 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 $\langle\omega; S, P\rangle$ 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.

Article information

Source
J. Symbolic Logic, Volume 58, Issue 2 (1993), 672-687.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744255

Mathematical Reviews number (MathSciNet)
MR1233932

Zentralblatt MATH identifier
0785.03002

JSTOR

Citation

Bateman, P. T.; Jockusch, C. G.; Woods, A. R. Decidability and Undecidability of Theories with a Predicate for the Primes. J. Symbolic Logic 58 (1993), no. 2, 672--687. https://projecteuclid.org/euclid.jsl/1183744255