Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T13:23:08.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Model-theoretic properties characterizing Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Richard Kaye
Richard Kaye*
Affiliation:
Jesus College, Oxford OX1 3DW, England
Get access Rights & Permissions [Opens in a new window]

Abstract

Let = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying

(i) K has no proper elementary substructures, and

(ii) whenever L ≻ K is a countable elementary extension there is and such that .

Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 949 - 963
Copyright
Copyright © Association for Symbolic Logic 1991

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

Barwise, J. and Schupf, J. [1976], An introduction to recursively saturated and resplendent models, this Journal, vol. 41, pp. 531–536.Google Scholar
Gaifman, H. [1972], A note on models and submodels of arithmetic, Conference in mathematical logic, London '70, Lecture Notes in Mathematics, vol. 255, Springer-Verlag, Berlin, pp. 128–144.CrossRefGoogle Scholar
Gaifman, H. and Dimitracopoulos, C. [1982], Fragments of Peano's arithmetic and the MRDP theorem, Logic and algorithmic, Monographies de l'Enseignement Mathématique, no. 50, Université de Genève, Geneva, pp. 187–206.Google Scholar
Kaye, R. [1987], Diophantine and parameter-free induction, Ph.D. thesis, Manchester University, Manchester.Google Scholar
Kaye, R.[1991a], Models of Peano arithmetic, Oxford University Press, Oxford.Google Scholar
Kaye, R.[1991b], On cofinal extensions of models of fragments of arithmetic, Notre Dame Journal of Formal Logic (to appear).Google Scholar
Kirby, L. and Paris, J. [1977], Initial segments of models of Peano's axioms, Set theory and hierarchy theory. V, Lecture Notes in Mathematics, vol. 619, Springer-Verlag, Berlin, pp. 211–226.CrossRefGoogle Scholar
Kossak, R. [1990], On extensions of models of strong fragments of arithmetic, Proceedings of the American Mathematical Society, vol. 108, pp. 223–232.CrossRefGoogle Scholar
Lessan, H. [1978], Models of Peano arithmetic, Ph.D. thesis, Manchester University, Manchester.Google Scholar
MacDowell, R. and Specker, E. [1961], Modelle der Arithmetik, Infinitistic methods, PWN, Warsaw, and Pergamon Press, Oxford, pp. 257–263.Google Scholar
Paris, J. [1980], A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic, Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, pp. 312–337.CrossRefGoogle Scholar
Paris, J. and Kirby, L. [1978], Σ n collection schémas in arithmetic, Logic colloquium '77, North-Holland, Amsterdam, pp. 199–209.CrossRefGoogle Scholar
Parsons, C. [1970], On a number theoretic choice schema and its relation to induction, Intuitionism and proof theory, North-Holland, Amsterdam, pp. 459–473.Google Scholar
Skolem, T. [1934], Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlvariablen, Fundament a Mathematicae, vol. 23, pp. 150–161.CrossRefGoogle Scholar
Wilkie, A. and Paris, J. [1989], On the existence of end extensions of models of bounded induction, Logic, methodology and philosophy of science. VIII, North-Holland, Amsterdam, pp. 143–161.Google Scholar