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Decidability and undecidability of theories with a predicate for the primes

Published online by Cambridge University Press:  12 March 2014

P. T. Bateman ,
C. G. Jockusch  and
A. R. Woods
P. T. Bateman
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois61801, E-mail: jockusch@math.uiuc.edu
C. G. Jockusch
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois61801, E-mail: jockusch@math.uiuc.edu
A. R. Woods
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, Western Australia 6009, Australia, E-mail: woods%maths.uwa.oz.qu@uunet.uu.net
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Abstract

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, 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 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 672 - 687
Copyright
Copyright © Association for Symbolic Logic 1993

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