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SURREAL TIME AND ULTRATASKS

Published online by Cambridge University Press:  30 August 2016

HAIDAR AL-DHALIMY  and
CHARLES J. GEYER
HAIDAR AL-DHALIMY*
Affiliation:
Department of Philosophy, University of Minnesota
CHARLES J. GEYER*
Affiliation:
School of Statistics, University of Minnesota
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF MINNESOTA 831 HELLER HALL 271 19TH AVENUE SOUTH MINNEAPOLIS, MN 55455, USA E-mail: haidar@umn.edu
†SCHOOL OF STATISTICS UNIVERSITY OF MINNESOTA 313 FORD HALL 224 CHURCH STREET SE MINNEAPOLIS, MN 55455, USA URL: users.stat.umn.edu/∼geyer E-mail: geyer@umn.edu
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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
Information
The Review of Symbolic Logic , Volume 9 , Issue 4 , December 2016 , pp. 836 - 847
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Alling, N. L. (1985). Conway’s field of surreal numbers. Transactions of the American Mathematical Society, 287(1), 365–386.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), 387–390.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.Google Scholar
Ehrlich, P. (2012). The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic, 18(1), 1–45.CrossRefGoogle Scholar
Feferman, S. (2009). Conceptions of the continuum. Intellectica, 51(1), 169–189.Google Scholar
Hellman, G., & Shapiro, S. (2013). The classical continuum without points. Review of Symbolic Logic, 6(3), 488–512.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, 1–39.CrossRefGoogle Scholar
Szabó, Z. G. (2010). Tasks and ultra-tasks. Hungarian Philosophical Review, 54(4), 177–190.Google Scholar
van den Dries, L., & Ehrlich, P. (2001). Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, 167(2), 173–188. 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. 401–431.Google Scholar