, Volume 190, Issue 1, pp 37–61 | Cite as

Fair infinite lotteries

  • Sylvia WenmackersEmail author
  • Leon Horsten
Open Access


This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.


Foundations of probability Non-standard analysis Countable additivity Infinity 



Thanks to Hannes Leitgeb, Vieri Benci, Richard Pettigrew, and Lieven Decock for valuable discussions and comments. The contribution of SW was supported by the Odysseus Grant Formal Epistemology: Foundations and Applications funded by the Research Foundation Flanders (FWO-Vlaanderen). The contribution of LH was in part supported by the AHRC project Foundations of Structuralism (AH/H001670/1).

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. Bartha P. (2004) Countable additivity and the de Finetti lottery. British Journal for the Philosophy of Science 55: 301–321CrossRefGoogle Scholar
  2. Benci V., Di Nasso M. (2003a) Alpha-theory: An elementary axiomatic for nonstandard analysis. Expositiones Mathematicae 21: 355–386CrossRefGoogle Scholar
  3. Benci V., Di Nasso M. (2003b) Numerosities of labelled sets: A new way of counting. Advances in Mathematics 173: 50–67CrossRefGoogle Scholar
  4. Benci V., Di Nasso M., Forti M. (2006) The eightfold path to nonstandard analysis. In: Cutland N. J., Di Nasso M., Ross D. A. (eds) Nonstandard methods and applications in mathematics, Lecture notes in logic. Association for Symbolic Logic, AK Peters, Helsinki, pp 3–44Google Scholar
  5. Benci, V., Galatolo, S. & Ghimenti, M. (2008). An elementary approach to stochastic differential equations using infinitesimals. Unpublished manuscript;
  6. Cutland N. (1983) Nonstandard measure theory and its applications. Bulletin of the London Mathematical Society 15: 529–589CrossRefGoogle Scholar
  7. de Finetti B. (1974) Theory of probability, Vols. 1 & 2 (A. Machí. Wiley, New YorkGoogle Scholar
  8. Douven I., Horsten L., Romeijn J.-W. (2010) Probabilist anti-realism. Pacific Philosophical Quarterly 91: 38–63CrossRefGoogle Scholar
  9. Dudley R. M. (2004) Real analysis and probability. Cambridge University Press, CambridgeGoogle Scholar
  10. Easwaran, K. (2010). Regularity and infinitesimal credences. Unpublished manuscript.Google Scholar
  11. Elga A. (2004) Infinitesimal chances and the laws of nature. Australasian Journal of Philosophy 82: 67–76CrossRefGoogle Scholar
  12. Hájek, A. (2010). Staying regular. Unpublished manuscript.Google Scholar
  13. Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitrechnung. (Ergebnisse Der Mathematik.) Foundations of probability. (2nd ed.) (N. Morrison, Trans., 1956). USA: Chelsea Publishing Company.Google Scholar
  14. Komjáth P., Totik V. (2008) Ultrafilters. American Mathematical Monthly 115: 33–44Google Scholar
  15. Lauwers, L. (2010). Purely finitely additive measures are non-constructible objects. Working paper, unpublished. Retrieved: May 29, 2010, from
  16. Lavine S. (1995) Finite mathematics. Synthese 103: 389–420CrossRefGoogle Scholar
  17. Lewis D. (1980) A subjectivist’s guide to objective chance. In: Jeffrey R. (eds) Studies in inductive logic and probability II. University of California Press, CaliforniaGoogle Scholar
  18. Lewis D. (1986) Philosophical papers. Oxford University Press, OxfordGoogle Scholar
  19. Mancosu P. (2009) Measuring the size of infinite collections of natural numbers. The Review of Symbolic Logic 2: 612–646CrossRefGoogle Scholar
  20. McCall S., Armstrong D. M. (1989) God’s lottery. Analysis 49: 223–224Google Scholar
  21. Ramsey F. P. (1926) Truth and probability. In: Braithwaite R. B. (eds) Foundations of mathematics and other essays. Routledge & P. Kegan, London, pp 156–198Google Scholar
  22. Robinson A. (1966) Non-standard analysis. North-Holland Publishing Company, New YorkGoogle Scholar
  23. Rubio J. E. (1994) Optimization and nonstandard analysis. Marcel Dekker, New YorkGoogle Scholar
  24. Schurz G., Leitgeb H. (2008) Finitistic and frequentistic approximation of probability measures with or without σ-additivity. Studia Logica 89: 257–283CrossRefGoogle Scholar
  25. Skyrms B. (1980) Causal Necessity. Yale University Press, New HavenGoogle Scholar
  26. Tenenbaum G. (1995) Introduction to analytic and probabilistic number theory. Cambridge University Press, CambridgeGoogle Scholar
  27. Truss J. (1997) Foundations of mathematical analysis. Clarendon Press, OxfordGoogle Scholar
  28. Väth M. (2007) Nonstandard analysis. Springer, BerlinGoogle Scholar
  29. Williamson T. (2007) How probable is an infinite sequence of heads?. Analysis 67: 173–180Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Faculty of PhilosophyUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of PhilosophyUniversity of BristolBristolUK

Personalised recommendations