, Volume 191, Issue 8, pp 1757–1760 | Cite as

Indeterminacy of fair infinite lotteries

  • Philip Kremer


In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers.


Foundations of probability Non-standard analysis Infinite lotteries 


  1. Kervliet, T. (2013). A uniform probability measure on the natural numbers. Master Thesis, Department of Mathematics, University of Amsterdam, Amsterdam.Google Scholar
  2. Wenmackers, S., & Horsten, L. (2013). Fair infinite lotteries. Synthese, 190, 37–61.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of TorontoTorontoCanada

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