Skip to main content Accessibility help
×
Home
Hostname: page-component-6c8bd87754-hvdfp Total loading time: 0.59 Render date: 2022-01-21T03:47:41.644Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

The Complexity of Classification Problems for Models of Arithmetic

Published online by Cambridge University Press:  15 January 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 1636.CrossRefGoogle 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. 894914.CrossRefGoogle Scholar
[4] Gaifman, Haim, Models and types of Peano's arithmetic, Annals of Mathematical Logic, vol. 9 (1976), no. 3, pp. 223306.CrossRefGoogle 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. 903928.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. 221272.CrossRefGoogle Scholar
[7] Kanovei, Vladimir, Borel equivalence relations: Structure and classification, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008.CrossRefGoogle 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. 6799.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. 573579.CrossRefGoogle 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. 9397 (electronic).CrossRefGoogle Scholar
[12] Smoryński, C., Recursively saturated nonstandard models of arithmetic, The Journal of Symbolic Logic, vol. 46 (1981), no. 2, pp. 259286.CrossRefGoogle Scholar
[13] Smoryński, C., Back-and-forth inside a recursively saturated model of arithmetic, Logic Colloquium '80 (Prague, 1980), North-Holland, Amsterdam, 1982, pp. 273278.Google Scholar
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The Complexity of Classification Problems for Models of Arithmetic
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The Complexity of Classification Problems for Models of Arithmetic
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The Complexity of Classification Problems for Models of Arithmetic
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *