Skip to main content Accessibility help
×
Home
Hostname: page-component-77ffc5d9c7-6tv98 Total loading time: 0.458 Render date: 2021-04-23T17:16:37.315Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Stability and posets

Published online by Cambridge University Press:  12 March 2014

Carl G. Jockusch Jr.
Affiliation:
University of Illinois, Department of Mathematics, Urbana, Il 61801-2975, USA, E-mail: jockusch@math.uiuc.edu
Bart Kastermans
Affiliation:
University of Wisconsin, Department of Mathematics, Madison, Wi 53706-1388, USA, E-mail: kasterma@math.wisc.edu, E-mail: lempp@math.wisc.edu
Steffen Lempp
Affiliation:
University of Wisconsin, Department of Mathematics, Madison, Wi 53706-1388, USA, E-mail: kasterma@math.wisc.edu, E-mail: lempp@math.wisc.edu
Manuel Lerman
Affiliation:
University of Connecticut, Department of Mathematics, Storrs, Ct 06269-3009, USA, E-mail: lerman@math.uconn.edu, E-mail: solomon@math.uconn.edu
Reed Solomon
Affiliation:
University of Connecticut, Department of Mathematics, Storrs, Ct 06269-3009, USA, E-mail: lerman@math.uconn.edu, E-mail: solomon@math.uconn.edu

Abstract

Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA0 to WSAC, the corresponding principle for weakly stable posets.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Cholak, Peter A., Jockusch, Carl G. Jr., and Slaman, Theodore A., On the strength of Ramsey's theorem for pairs, this Journal, vol. 66 (2001), pp. 155.Google Scholar
[2]Demuth, Oskar and Kučera, Antonín, Remarks on l-genericity, semigenericity, and related concepts, Commentationes Mathematicae Universitatis Carolinae, vol. 28 (1987), pp. 8594.Google Scholar
[3]Harizanov, Valentina S., Jockusch, Carl G. Jr., and Knight, Julia F., Chains and antichains in partial orderings, Archive for Mathematical Logic, vol. 48 (2009), pp. 3953.CrossRefGoogle Scholar
[4]Herrmann, Eberhard, Infinite chains and antichains in computable partial orderings, this Journal, vol. 66 (2001), pp. 923934.Google Scholar
[5]Hirschfeldt, Denis R., Jockusch, Carl G. Jr, Kjos-Hanssen, Bjørn, Lempp, Steffen, and Slaman, Theodore A., The strength of some combinatorial principles related to Ramsey's theorem for pairs, Proceedings of the Program on Computational Prospects of Infinity (Chong, C. T., Feng, Qi, Slaman, Theodore, Woodin, Hugh, and Yang, Yue, editors), IMS Proceedings, World Scientific, 2007, pp. 143161.Google Scholar
[6]Hirschfeldt, Denis R. and Shore, Richard A., >Combinatorial principles weaker than Ramsey's theorem for pairs, this Journal, vol. 72 (2007), pp. 171206.Combinatorial+principles+weaker+than+Ramsey's+theorem+for+pairs,+this+Journal,+vol.+72+(2007),+pp.+171–206.>Google Scholar
[7]Hummel, Tamara L., Effective versions of Ramsey's theorem: Avoiding the cone above 0′, this Journal, vol. 59 (1994), pp. 13011325.Google Scholar
[8]Hummel, Tamara L. and Jockusch, Carl G. Jr., Generalized cohesiveness, this Journal, vol. 64 (1999), pp. 489516.Google Scholar
[9]Jockusch, Carl G. Jr., Ramsey's theorem and recursion theory, this Journal, vol. 37 (1972), pp. 268280.Google Scholar
[10]Rogers, Hartley Jr, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York-Toronto-London, 1967.Google Scholar
[11]Simpson, Stephen G., Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, Heidelberg, 1999.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 13 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 23rd April 2021. This data will be updated every 24 hours.

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.

Stability and posets
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.

Stability and posets
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.

Stability and posets
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *