Automorphisms of Countable Short Recursively Saturated Models of PA
Erez Shochat
Source: Notre Dame J. Formal Logic Volume 49, Number 4
(2008), 345-360.
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 which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model.
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Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1224257535
Digital Object Identifier: doi:10.1215/00294527-2008-016
Mathematical Reviews number (MathSciNet): MR2456652
Zentralblatt MATH identifier: 05657672
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.
Mathematical Reviews (MathSciNet): MR0409172
Zentralblatt MATH: 0343.02031
[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.
Mathematical Reviews (MathSciNet): MR1656934
Zentralblatt MATH: 0920.03069
[3] Blass, A., "The intersection of nonstandard models of arithmetic", The Journal of Symbolic Logic, vol. 37 (1972), pp. 103--6.
Mathematical Reviews (MathSciNet): MR0323560
Zentralblatt MATH: 0246.02039
Digital Object Identifier: doi:10.2307/2272552
JSTOR: links.jstor.org
[4] Ehrenfeucht, A., "Discernible elements in models for Peano Arithmetic", The Journal of Symbolic Logic, vol. 38 (1973), pp. 291--92.
Mathematical Reviews (MathSciNet): MR0337583
Zentralblatt MATH: 0279.02036
Digital Object Identifier: doi:10.2307/2272063
JSTOR: links.jstor.org
[5] Gaifman, H., "Models and types of Peano's arithmetic", Annals of Pure and Applied Logic, vol. 9 (1976), pp. 223--306.
Mathematical Reviews (MathSciNet): MR0406791
Zentralblatt MATH: 0332.02058
[6] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR1221741
Zentralblatt MATH: 0789.03031
[7] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
Mathematical Reviews (MathSciNet): MR1098499
Zentralblatt MATH: 0744.03037
[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.
Mathematical Reviews (MathSciNet): MR1325480
Zentralblatt MATH: 0824.03015
[9] Kaye, R., and D. Macpherson, editors, Automorphisms of First-Order Structures, Oxford Science Publications, The Clarendon Press, New York, 1994.
Mathematical Reviews (MathSciNet): MR1325468
Zentralblatt MATH: 0797.00010
[10] Kossak, R., "A certain class of models of Peano Arithmetic", The Journal of Symbolic Logic, vol. 48 (1983), pp. 311--20.
Mathematical Reviews (MathSciNet): MR84j:03076
Zentralblatt MATH: 0514.03036
Digital Object Identifier: doi:10.2307/2273548
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet): MR1462201
Zentralblatt MATH: 0873.03037
Digital Object Identifier: doi:10.1007/s001530050056
[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.
Mathematical Reviews (MathSciNet): MR1257467
Zentralblatt MATH: 0796.03043
Digital Object Identifier: doi:10.1016/0168-0072(93)90035-C
[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.
Mathematical Reviews (MathSciNet): MR2250469
Zentralblatt MATH: 1101.03029
[14] Kotlarski, H., ``On elementary cuts in models of arithmetic,'' Fundamenta Mathematicae, vol. 115 (1983), pp. 27--31.
Mathematical Reviews (MathSciNet): MR690667
Zentralblatt MATH: 0515.03038
[15] Kotlarski, H., "On elementary cuts in recursively saturated models of Peano Arithmetic", Fundamenta Mathematicae, vol. 120 (1984), pp. 205--22.
Mathematical Reviews (MathSciNet): MR755777
Zentralblatt MATH: 0572.03016
[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.
Mathematical Reviews (MathSciNet): MR0462940
Zentralblatt MATH: 0316.02018
Digital Object Identifier: doi:10.1007/BFb0081120
[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.
Mathematical Reviews (MathSciNet): MR2210062
Zentralblatt MATH: 1101.03030
Digital Object Identifier: doi:10.2178/jsl/1140641169
Project Euclid: euclid.jsl/1140641169
[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.
Mathematical Reviews (MathSciNet): MR0465849
Zentralblatt MATH: 0376.02032
[20] Schmerl, J. H., "Automorphism groups of models of Peano Arithmetic", The Journal of Symbolic Logic, vol. 67 (2002), pp. 1249--64.
Mathematical Reviews (MathSciNet): MR1955236
Zentralblatt MATH: 1038.03044
Digital Object Identifier: doi:10.2178/jsl/1190150283
Project Euclid: euclid.jsl/1190150283
[21] Schmerl, J. H., "Moving intersticial gaps", Mathematical Logic Quarterly, vol. 48 (2002), pp. 283--96.
Mathematical Reviews (MathSciNet): MR1883243
Zentralblatt MATH: 0997.03036
Digital Object Identifier: doi:10.1002/1521-3870(200202)48:2<283::AID-MALQ283>3.0.CO;2-%23
[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.
Mathematical Reviews (MathSciNet): MR0613281
Zentralblatt MATH: 0501.03044
Digital Object Identifier: doi:10.2307/2273620
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet): MR1155392
Zentralblatt MATH: 0743.03029
Digital Object Identifier: doi:10.1002/malq.19910371304
[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.
Mathematical Reviews (MathSciNet): MR606795
Zentralblatt MATH: 0463.03017
Digital Object Identifier: doi:10.1007/BFb0090175
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