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The Complexity of Classification Problems for Models of Arithmetic

Published online by Cambridge University Press:  15 January 2014

Samuel Coskey  and
Roman Kossak
Samuel Coskey
Affiliation:
Mathematics Program, The Cuny Graduate Center, 365 Fifth Avenue, New York, NY 10016-4309, USA, E-mail: scoskey@nylogic.org, E-mail: rkossak@nylogic.org
Roman Kossak
Affiliation:
Mathematics Program, The Cuny Graduate Center, 365 Fifth Avenue, New York, NY 10016-4309, USA, E-mail: scoskey@nylogic.org, E-mail: rkossak@nylogic.org
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Abstract

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 3 , September 2010 , pp. 345 - 358
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1] Coskey, Samuel, Ellis, Paul, and Schneider, Scott, The conjugacy problem for the automorphism group of the random graph, 2009, submitted and available at arXiv:0902.4038v2 [math.L0].Google Scholar
[2] Enayat, Ali, Automorphisms of models of arithmetic: a unified view, Annals of Pure and Applied Logic, vol. 145 (2007), no. 1, pp. 16–36.Google Scholar
[3] Friedman, Harvey and Stanley, Lee, A Borel reducibility theory for classes of countable structures, The Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 894–914.CrossRefGoogle Scholar
[4] Gaifman, Haim, Models and types of Peano's arithmetic, Annals of Mathematical Logic, vol. 9 (1976), no. 3, pp. 223–306.Google Scholar
[5] Harrington, Leo A., Kechris, Alexander S., and Louveau, Alain, Borel equivalence relations and classifications of countable models, Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903–928.CrossRefGoogle Scholar
[6] Hjorth, Greg and Kechris, Alexander S., Borel equivalence relations and classifications of countable models, Annals of Pure and Applied Logic, vol. 82 (1996), no. 3, pp. 221–272.Google Scholar
[7] Kanovei, Vladimir, Borel equivalence relations: Structure and classification, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008.Google Scholar
[8] Kaye, Richard, Kossak, Roman, and Kotlarski, Henryk, Automorphisms of recursively saturated models of arithmetic, Annals of Pure and Applied Logic, vol. 55 (1991), no. 1, pp. 67–99.CrossRefGoogle Scholar
[9] Kossak, Roman and Schmerl, James H., The structure of models of Peano arithmetic, Oxford Logic Guides, vol. 50, The Clarendon Press Oxford University Press, Oxford, 2006.CrossRefGoogle Scholar
[10] Kotlarski, Henryk, Full satisfaction classes: a survey, Notre Dame Journal of Formal Logic, vol. 32 (1991), no. 4, pp. 573–579.Google Scholar
[11] Marker, David, The Borel complexity of isomorphism for theories with many types, Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 93–97 (electronic).Google Scholar
[12] Smoryński, C., Recursively saturated nonstandard models of arithmetic, The Journal of Symbolic Logic, vol. 46 (1981), no. 2, pp. 259–286.Google Scholar
[13] Smoryński, C., Back-and-forth inside a recursively saturated model of arithmetic, Logic Colloquium '80 (Prague, 1980), North-Holland, Amsterdam, 1982, pp. 273–278.Google Scholar