Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-j5sqr Total loading time: 0.289 Render date: 2022-10-01T06:15:30.325Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

ON THE STRENGTH OF TWO RECURRENCE THEOREMS

Published online by Cambridge University Press:  29 September 2016

ADAM R. DAY*
Affiliation:
SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALANDE-mail: adam.day@vuw.ac.nz

Abstract

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between WKL and ACA (working over RCA0). This is the first example of a theorem with this property. It also shows the existence of an almost periodic point is conservative over RCA0 for ${\rm{\Pi }}_1^1$-sentences.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

REFERENCES

Beiglböck, M. and Towsner, H., Transfinite approximation of Hindman’s theorem . Israel Journal of Mathematics, vol. 191 (2012), no. 1, pp. 4159.CrossRefGoogle Scholar
Blass, A. R., Hirst, J. L., and Simpson, S. G., Logical analysis of some theorems of combinatorics and topological dynamics. Logic and combinatorics (Arcata, Calif., 1985), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 125–156.
Friedman, H., Simpson, S. G., and Yu, X., Periodic points and subsystems of second-order arithmetic . Annals of Pure and Applied Logic, vol. 62 (1993), no. 1, pp. 5164. Logic Colloquium ’89 (Berlin).Google Scholar
Montalbán, A., Open questions in reverse mathematics . Bulletin of Symbolic Logic, vol. 17 (2011), no. 3, pp. 431454.CrossRefGoogle Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic. 2nd edition, Cambridge University Press, New York, 2009.CrossRefGoogle Scholar
Tao, T., Poincaré’s Legacies, Part I. American Mathematical Society, Providence, 2009.Google Scholar
Towsner, H., A combinatorial proof of the dense Hindman’s theorem . Discrete Mathematics, vol. 311 (2011), no. 14, pp. 13801384.CrossRefGoogle Scholar
Towsner, H., Hindman’s theorem: an ultrafilter argument in second order arithmetic, this Journal, vol. 76 (2011), no. 1, pp. 353360.
Towsner, H., A simple proof and some difficult examples for Hindman’s theorem . Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 1, pp. 5365.CrossRefGoogle Scholar
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@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 saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved 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.

ON THE STRENGTH OF TWO RECURRENCE THEOREMS
Available formats
×

Save article to Dropbox

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

ON THE STRENGTH OF TWO RECURRENCE THEOREMS
Available formats
×

Save article to Google Drive

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

ON THE STRENGTH OF TWO RECURRENCE THEOREMS
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? *