Decidability and undecidability of theories with a predicate for the primes

Journal of Symbolic Logic 58 (2):672-687 (1993)
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
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DOI 10.2307/2275227
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J. Richard Büchi (1960). Weak Second-Order Arithmetic and Finite Automata. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 6 (1-6):66-92.

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