Notre Dame Journal of Formal Logic

Automorphisms of Saturated and Boundedly Saturated Models of Arithmetic

Ermek S. Nurkhaidarov and Erez Shochat

Abstract

We discuss automorphisms of saturated models of PA and boundedly saturated models of PA. We show that Smoryński's Lemma and Kaye's Theorem are not only true for countable recursively saturated models of PA but also true for all boundedly saturated models of PA with slight modifications.

Article information

Source
Notre Dame J. Formal Logic, Volume 52, Number 3 (2011), 315-329.

Dates
First available in Project Euclid: 28 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1311875777

Digital Object Identifier
doi:10.1215/00294527-1435483

Mathematical Reviews number (MathSciNet)
MR2822492

Zentralblatt MATH identifier
1241.03046

Subjects
Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx]

Keywords
saturation bounded saturation automorphism groups models of PA

Citation

Nurkhaidarov, Ermek S.; Shochat, Erez. Automorphisms of Saturated and Boundedly Saturated Models of Arithmetic. Notre Dame J. Formal Logic 52 (2011), no. 3, 315--329. doi:10.1215/00294527-1435483. https://projecteuclid.org/euclid.ndjfl/1311875777


Export citation

References

  • [1] Blass, A., "The intersection of nonstandard models of arithmetic", The Journal of Symbolic Logic, vol. 37 (1972), pp. 103–6.
  • [2] Gaifman, H., "Models and types of Peano's arithmetic", Annals of Pure and Applied Logic, vol. 9 (1976), pp. 223–306.
  • [3] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
  • [4] Kaye, R., "A Galois correspondence for countable recursively saturated models of Peano Arithmetic", pp. 293–312 in Automorphisms of First-Order Structures, Oxford Science Publications, Oxford University Press, New York, 1994.
  • [5] Kaye, R., R. Kossak, and H. Kotlarski, "Automorphisms of recursively saturated models of arithmetic", Annals of Pure and Applied Logic, vol. 55 (1991), pp. 67–99.
  • [6] Kossak, R., H. Kotlarski, and J. H. Schmerl, "On maximal subgroups of the automorphism group of a countable recursively saturated model of PA", Annals of Pure and Applied Logic, vol. 65 (1993), pp. 125–48.
  • [7] Kossak, R., and J. H. Schmerl, The Structure of Models of Peano Arithmetic, vol. 50 of Oxford Logic Guides, The Clarendon Press, New York, 2006.
  • [8] Kotlarski, H., "On elementary cuts in recursively saturated models of Peano Arithmetic", Fundamenta Mathematicae, vol. 120 (1984), pp. 205–22.
  • [9] Nurkhaidarov, E. S., "Automorphism groups of saturated models of PA" of cardinality $\aleph_1$, Collected Works Devoted to the Memory of A. D. Taimanov, (2006), pp. 295–97.
  • [10] Pabion, J.-F., "Saturated models of Peano arithmetic", The Journal of Symbolic Logic, vol. 47 (1982), pp. 625–37.
  • [11] Schmerl, J. H., "Closed normal subgroups", Mathematical Logic Quarterly, vol. 47 (2001), pp. 489–92.
  • [12] Shochat, E., "Automorphisms of countable short recursively saturated models of PA", Notre Dame Journal of Formal Logic, vol. 49 (2008), pp. 345–60.
  • [13] Smoryński, C., "Back-and-forth inside a recursively saturated model of arithmetic", pp. 273–78 in Logic Colloquium '80 (Prague, 1980), vol. 108 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1982.