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SURREAL TIME AND ULTRATASKS

  • HAIDAR AL-DHALIMY (a1) and CHARLES J. GEYER (a2)

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.

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Corresponding author

*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

References

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Alling, N. L. (1985). Conway’s field of surreal numbers. Transactions of the American Mathematical Society, 287(1), 365386.
Bell, J. L. (2008). A Primer of Infinitesimal Analysis (second edition). Cambridge University Press, New York.
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/.
Clark, P., & Read, S. (1984). Hypertasks. Synthese, 61(3), 387390.
Conway, J. H. (2001). On Numbers and Games (second edition). Natick, MA: A. K. Peters.
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.
Feferman, S. (2009). Conceptions of the continuum. Intellectica, 51(1), 169189.
Hellman, G., & Shapiro, S. (2013). The classical continuum without points. Review of Symbolic Logic, 6(3), 488512.
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.
Lewis, D. K. (1991). Parts of Classes. Oxford: Blackwell.
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.
Rubinstein-Salzedo, S. & Swaminathan, A. (2014). Analysis on surreal numbers. Journal of Logic & Analysis, 6, 139.
Szabó, Z. G. (2010). Tasks and ultra-tasks. Hungarian Philosophical Review, 54(4), 177190.
van den Dries, L., & Ehrlich, P. (2001). Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, 167(2), 173188. Erratum, 168(3), 295–297.
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.
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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