Decidability and classification of the theory of integers with primes

Itay Kaplan, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We show that under Dickson's conjecture about the distribution of primes in the natural numbers, the theory Th (+, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (+, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

Original languageAmerican English
Pages (from-to)1041-1050
Number of pages10
JournalJournal of Symbolic Logic
Issue number3
StatePublished - 1 Sep 2017

Bibliographical note

Funding Information:
§4. Acknowledgements. The first author would like to thank the Israel Science foundation for partial support of this research (Grant no. 1533/14). The research leading to these results has received funding from the European Research Council, ERC Grant Agreement n. 338821. No. 1082 on the second author’s list of publications.

Publisher Copyright:
Copyright © The Association for Symbolic Logic 2017.


  • Dickson's conjecture
  • decidability
  • model theory
  • primes


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