Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T07:26:40.844Z Has data issue: false hasContentIssue false
  • English
  • Français

The laws of integer divisibility, and solution sets of linear divisibility conditions

Published online by Cambridge University Press:  12 March 2014

L. van den Dries  and
A. J. Wilkie
L. van den Dries
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA, E-mail: [email protected]
A. J. Wilkie
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, UK, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 503 - 526
Copyright
Copyright © Association for Symbolic Logic 2003

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

[1]Bel'tyukov, A. P.. Decidability of the universal theory of natural numbers with + and ∣, Seminars of Steklov Mathematical Institute (Leningrad), vol. 60 (1976), pp. 15–28.Google Scholar
[2]van den Dries, L., Quantifier elimination for linear formulas over ordered and valued fields. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981). pp. 19–33.Google Scholar
[3]Groemer, H., On the extension of additive functionals on classes of convex sets, Pacific Journal of Mathematics, vol. 75 (1978), pp. 397–410.CrossRefGoogle Scholar
[4]Lipshitz, L., The diophantine problem for addition and divisibility, Transactions of the American Mathematical Society, vol. 235 (1978), pp. 271–283.CrossRefGoogle Scholar
[5]Lipshitz, L., Some remarks on the diophantine problem for addition and divisibility. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981), pp. 41–52.Google Scholar
[6]Macintyre, A., A theorem of Rabin in a general setting. Bulletin de la Société Mathématique de Belgique, vol. 33 (1981), pp. 53–63.Google Scholar
[7]Moschovakis, Y., On primitive recursive algorithms and the greatest common divisor function, Theoretical Computer Science, to appear.Google Scholar
[8]Wilkie, A.J., Modèles non standard de l'arithmétique, et complexité algorithmique. Modèles non standard en arithmétique et théorie des ensembles. Publications Mathématiques de l'Université Paris VII, 1984, pp. 5–45.Google Scholar