## 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]

#### 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

#### 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.