Automorphisms of Saturated and Boundedly Saturated Models of Arithmetic
Ermek S. Nurkhaidarov and Erez Shochat
Source: Notre Dame J. Formal Logic Volume 52, Number 3
(2011), 315-329.
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.
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Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1311875777
Digital Object Identifier: doi:10.1215/00294527-1435483
Mathematical Reviews number (MathSciNet): MR2822492
Zentralblatt MATH identifier: 05970097
References
[1] 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
[2] 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
[3] 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
[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.
Mathematical Reviews (MathSciNet): MR1325480
Zentralblatt MATH: 0824.03015
[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.
Mathematical Reviews (MathSciNet): MR1134917
Zentralblatt MATH: 0748.03023
Digital Object Identifier: doi:10.1016/0168-0072(91)90098-7
[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.
Mathematical Reviews (MathSciNet): MR1257467
Zentralblatt MATH: 0796.03043
Digital Object Identifier: doi:10.1016/0168-0072(93)90035-C
[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.
Mathematical Reviews (MathSciNet): MR2250469
Zentralblatt MATH: 1101.03029
[8] 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
[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.
Mathematical Reviews (MathSciNet): MR666820
Zentralblatt MATH: 0498.03021
Digital Object Identifier: doi:10.2307/2273592
[11] Schmerl, J. H., "Closed normal subgroups", Mathematical Logic Quarterly, vol. 47 (2001), pp. 489–92.
Mathematical Reviews (MathSciNet): MR1865768
Zentralblatt MATH: 0997.03035
Digital Object Identifier: doi:10.1002/1521-3870(200111)47:4<489::AID-MALQ489>3.0.CO;2-8
[12] Shochat, E., "Automorphisms of countable short recursively saturated models of PA", Notre Dame Journal of Formal Logic, vol. 49 (2008), pp. 345–60.
Mathematical Reviews (MathSciNet): MR2456652
Zentralblatt MATH: 1185.03065
Digital Object Identifier: doi:10.1215/00294527-2008-016
Project Euclid: euclid.ndjfl/1224257535
[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.
Mathematical Reviews (MathSciNet): MR673798
Zentralblatt MATH: 0503.03033
Notre Dame Journal of Formal Logic