Notre Dame Journal of Formal Logic

Automorphisms of Countable Short Recursively Saturated Models of PA

Erez Shochat

Abstract

A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of PA are exactly the elementary initial segments of recursively saturated models of PA. In this paper, we survey and prove results on short recursively saturated models of PA and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted G |M(a) , contains all the automorphisms of a countable short recursively saturated model of PA which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model.

Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 4 (2008), 345-360.

Dates
First available in Project Euclid: 17 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1224257535

Digital Object Identifier
doi:10.1215/00294527-2008-016

Mathematical Reviews number (MathSciNet)
MR2456652

Zentralblatt MATH identifier
1185.03065

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

Keywords
short recursive saturation recursive saturation models of PA automorphisms

Citation

Shochat, Erez. Automorphisms of Countable Short Recursively Saturated Models of PA. Notre Dame J. Formal Logic 49 (2008), no. 4, 345--360. doi:10.1215/00294527-2008-016. https://projecteuclid.org/euclid.ndjfl/1224257535


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References

  • [1] Barwise, J., and J. Schlipf, ``On recursively saturated models of arithmetic,'' pp. 42--55 in Model Theory and Algebra (A Memorial Tribute to Abraham Robinson), vol. 498 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
  • [2] Bigorajska, T., H. Kotlarski, and J. H. Schmerl, "On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic", Fundamenta Mathematicae, vol. 158 (1998), pp. 125--46.
  • [3] Blass, A., "The intersection of nonstandard models of arithmetic", The Journal of Symbolic Logic, vol. 37 (1972), pp. 103--6.
  • [4] Ehrenfeucht, A., "Discernible elements in models for Peano Arithmetic", The Journal of Symbolic Logic, vol. 38 (1973), pp. 291--92.
  • [5] Gaifman, H., "Models and types of Peano's arithmetic", Annals of Pure and Applied Logic, vol. 9 (1976), pp. 223--306.
  • [6] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
  • [7] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
  • [8] 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.
  • [9] Kaye, R., and D. Macpherson, editors, Automorphisms of First-Order Structures, Oxford Science Publications, The Clarendon Press, New York, 1994.
  • [10] Kossak, R., "A certain class of models of Peano Arithmetic", The Journal of Symbolic Logic, vol. 48 (1983), pp. 311--20.
  • [11] Kossak, R., and N. Bamber, "On two questions concerning the automorphism groups of countable recursively saturated models of PA", Archive for Mathematical Logic, vol. 36 (1996), pp. 73--79.
  • [12] 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.
  • [13] Kossak, R., and J. H. Schmerl, The Structure of Models of Peano Arithmetic, vol. 50 of Oxford Logic Guides, The Clarendon Press, Oxford, 2006.
  • [14] Kotlarski, H., ``On elementary cuts in models of arithmetic,'' Fundamenta Mathematicae, vol. 115 (1983), pp. 27--31.
  • [15] Kotlarski, H., "On elementary cuts in recursively saturated models of Peano Arithmetic", Fundamenta Mathematicae, vol. 120 (1984), pp. 205--22.
  • [16] Kueker, D. W., "Back-and-forth arguments and infinitary logics", pp. 17--71 in Infinitary Logic: In Memoriam Carol Karp, vol. 492 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
  • [17] Lesan, H., Models of Arithmetic, Dissertation, University of Manchester, 1978.
  • [18] Nurkhaidarov, E. S., "Automorphism groups of arithmetically saturated models", The Journal of Symbolic Logic, vol. 71 (2006), pp. 203--16.
  • [19] Ressayre, J. P., "Models with compactness properties relative to an admissible language", Annals of Pure and Applied Logic, vol. 11 (1977), pp. 31--55.
  • [20] Schmerl, J. H., "Automorphism groups of models of Peano Arithmetic", The Journal of Symbolic Logic, vol. 67 (2002), pp. 1249--64.
  • [21] Schmerl, J. H., "Moving intersticial gaps", Mathematical Logic Quarterly, vol. 48 (2002), pp. 283--96.
  • [22] Shochat, E., Countable Short Recursively Saturated Models of Arithmetic, Dissertation, City University of New York, 2006.
  • [23] Smoryński, C., "Recursively saturated nonstandard models of arithmetic", The Journal of Symbolic Logic, vol. 46 (1981), pp. 259--86.
  • [24] Tzouvaras, A., "A note on real subsets of a recursively saturated model", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 37 (1991), pp. 207--16.
  • [25] Wilmers, G., ``Minimally saturated models,'' pp. 370--80 in Model Theory of Algebra and Arithmetic (Proceedings of the Conference, Karpacz, 1979), vol. 834 of Lecture Notes in Mathematics, Springer, Berlin, 1980.