Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T20:58:01.126Z Has data issue: false hasContentIssue false
  • English
  • Français

DECIDABILITY AND CLASSIFICATION OF THE THEORY OF INTEGERS WITH PRIMES

Published online by Cambridge University Press:  08 September 2017

ITAY KAPLAN  and
SAHARON SHELAH
ITAY KAPLAN
Affiliation:
THE HEBREW UNIVERSITY OF JERUSALEM EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS, GIVAT RAM JERUSALEM91904, ISRAELE-mail: [email protected]: https://sites.google.com/site/itay80/
SAHARON SHELAH
Affiliation:
THE HEBREW UNIVERSITY OF JERUSALEM EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS, GIVAT RAM JERUSALEM 91904, ISRAEL and DEPARTMENT OF MATHEMATICS HILL CENTER-BUSCH CAMPUS RUTGERS, THE STATE UNIVERSITY OF NEW JERSEY 110 FRELINGHUYSEN ROAD PISCATAWAY, NJ08854-8019, USA E-mail: [email protected]: http://shelah.logic.at/
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 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 (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

Keywords

03C4503F3003B2511A41model theorydecidabilityprimesDickson’s conjecture
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 1041 - 1050
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bateman, P. T., Jockusch, C. G., and Woods, A. R., Decidability and undecidability of theories with a predicate for the primes , this Journal, vol. 58 (1993), no. 2, pp. 672–687.Google Scholar
Bès, A., A survey of arithmetical definability . Bulletin of The Belgian Mathematical Society-Simon Stevin, 2001, Suppl., pp. 1–54.Google Scholar
Chatzidakis, Z. and Pillay, A., Generic structures and simple theories . Annals of Pure and Applied Logic, vol. 95 (1998), no. 1–3, pp. 71–92.Google Scholar
Chernikov, A., Palacin, D., and Takeuchi, K., On n-dependence . Notre Dame Journal of Formal Logic, 2014, accepted, arXiv:1411.0120.Google Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: Basic facts and examples. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.Google Scholar
Dickson, L. E., A new extension of Dirichlet’s theorem on prime numbers . Messenger of Mathematics, vol. 33 (1904), pp. 155–161.Google Scholar
Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions . Annals of Mathematics (2), vol. 167 (2008), no. 2, pp. 481–547.CrossRefGoogle Scholar
Korec, I., A list of arithmetical structures complete with respect to the first-order definability . Theoretical Computer Science, vol. 257 (2001), no. 1–2, pp. 115–151.CrossRefGoogle Scholar
Marker, D., Model Theory: An Introduction , Graduate Texts in Mathematics, vol. 217, Springer, New York, 2002.Google Scholar
Onshuus, A. and Usvyatsov, A., On dp-minimality, strong dependence and weight , this Journal, vol. 76 (2011), no. 3, pp. 737–758.Google Scholar
Palacin, D. and Sklinos, R., On superstable expansions of free abelian groups . Notre Dame Journal of Formal Logic, 2014, accepted, arXiv:1405.0568.Google Scholar
Poizat, B., Supergénérix . Journal of Algebra, vol. 404 (2014), pp. 240–270.CrossRefGoogle Scholar
Ribenboim, P., The Book of Prime Number Records, second ed., Springer-Verlag, New York, 1989.Google Scholar
Shelah, S., Classification Theory and the Number of Nonisomorphic Models, second ed. Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
Simon, P., On dp-minimal ordered structures , this Journal, vol. 76 (2011), no. 2, pp. 448–460.Google Scholar
Simon, P., A Guide to NIP Theories, Lecture Notes in Logic, vol. 44, Cambridge University Press, Cambridge, 2015.Google Scholar
Tao, T., Every odd number greater than 1 is the sum of at most five primes . Mathematics of Computation, vol. 83 (2013), no. 286, pp. 997–1038.Google Scholar
Tent, K. and Ziegler, M., A Course in Model Theory, Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012.Google Scholar
Woods, A. R., Some problems in logic and number theory, and their connections , New Studies in Weak Arithmetics (Cégielski, P., Cornaros, C., and Dimitracopoulos, C., editors), CSLI Lecture Notes, vol. 211, CSLI Publications, Stanford, CA, 2013, pp. 271–388. Dissertation, University of Manchester, Manchester, 1981.Google Scholar