Skip to main content Accessibility help
×
Home
Hostname: page-component-79b67bcb76-b5nxq Total loading time: 0.252 Render date: 2021-05-16T03:55:15.736Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

SURREAL TIME AND ULTRATASKS

Published online by Cambridge University Press:  30 August 2016

HAIDAR AL-DHALIMY
Affiliation:
Department of Philosophy, University of Minnesota
CHARLES J. GEYER
Affiliation:
School of Statistics, University of Minnesota
Corresponding
E-mail address:

Abstract

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2016 

Access options

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

References

Alling, N. L. (1985). Conway’s field of surreal numbers. Transactions of the American Mathematical Society, 287(1), 365386.Google Scholar
Bell, J. L. (2008). A Primer of Infinitesimal Analysis (second edition). Cambridge University Press, New York.CrossRefGoogle Scholar
Bell, J. L. (2014). Continuity and infinitesimals. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Winter 2014 edition). Available at: http://plato.stanford.edu/archives/win2014/entries/continuity/.Google Scholar
Clark, P., & Read, S. (1984). Hypertasks. Synthese, 61(3), 387390.CrossRefGoogle Scholar
Conway, J. H. (2001). On Numbers and Games (second edition). Natick, MA: A. K. Peters.Google Scholar
Costin, O., Ehrlich, P., & Friedman, H. M. (preprint). Integration on the surreals: A conjecture of Conway, Kruskal and Norton, submitted. Available at: http://arxiv.org/abs/1505.02478.
Ehrlich, P. (2012). The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic, 18(1), 145.CrossRefGoogle Scholar
Feferman, S. (2009). Conceptions of the continuum. Intellectica, 51(1), 169189.Google Scholar
Hellman, G., & Shapiro, S. (2013). The classical continuum without points. Review of Symbolic Logic, 6(3), 488512.CrossRefGoogle Scholar
Laraudogoitia, J. P. (2013). Supertasks. In Zalta, E. N. (editor). Stanford Encyclopedia of Philosophy (Fall 2013 edition). Available at: http://plato.stanford.edu/archives/fall2013/entries/spacetime-supertasks.Google Scholar
Lewis, D. K. (1991). Parts of Classes. Oxford: Blackwell.Google Scholar
Reck, E. (2012). Dedekind’s contributions to the foundations of mathematics. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Winter 2012 edition). Available at: http://plato.stanford.edu/archives/win2012/entries/dedekind-foundations.Google Scholar
Rubinstein-Salzedo, S. & Swaminathan, A. (2014). Analysis on surreal numbers. Journal of Logic & Analysis, 6, 139.CrossRefGoogle Scholar
Szabó, Z. G. (2010). Tasks and ultra-tasks. Hungarian Philosophical Review, 54(4), 177190.Google Scholar
van den Dries, L., & Ehrlich, P. (2001). Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, 167(2), 173188. Erratum, 168(3), 295–297.CrossRefGoogle Scholar
Zermelo, E. (1930). On boundary numbers and domains of sets: New investigations in the foundations of set theory. In Ebbinghaus, H.-D., Fraser, C. G., and Kanamori, A., editors. Ersnt Zermelo: Collected Works, Vol. I. Springer, Berlin, 2010, pp. 401431.Google Scholar

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.

SURREAL TIME AND ULTRATASKS
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.

SURREAL TIME AND ULTRATASKS
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.

SURREAL TIME AND ULTRATASKS
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *