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Fair infinite lotteries

Abstract

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.

References

  • Bartha P. (2004) Countable additivity and the de Finetti lottery. British Journal for the Philosophy of Science 55: 301–321

    Article  Google Scholar 

  • Benci V., Di Nasso M. (2003a) Alpha-theory: An elementary axiomatic for nonstandard analysis. Expositiones Mathematicae 21: 355–386

    Article  Google Scholar 

  • Benci V., Di Nasso M. (2003b) Numerosities of labelled sets: A new way of counting. Advances in Mathematics 173: 50–67

    Article  Google Scholar 

  • 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 3–44

    Google Scholar 

  • Benci, V., Galatolo, S. & Ghimenti, M. (2008). An elementary approach to stochastic differential equations using infinitesimals. Unpublished manuscript; http://arxiv.org/abs/0807.3477.

  • Cutland N. (1983) Nonstandard measure theory and its applications. Bulletin of the London Mathematical Society 15: 529–589

    Article  Google Scholar 

  • de Finetti B. (1974) Theory of probability, Vols. 1 & 2 (A. Machí. Wiley, New York

    Google Scholar 

  • Douven I., Horsten L., Romeijn J.-W. (2010) Probabilist anti-realism. Pacific Philosophical Quarterly 91: 38–63

    Article  Google Scholar 

  • Dudley R. M. (2004) Real analysis and probability. Cambridge University Press, Cambridge

    Google Scholar 

  • Easwaran, K. (2010). Regularity and infinitesimal credences. Unpublished manuscript.

  • Elga A. (2004) Infinitesimal chances and the laws of nature. Australasian Journal of Philosophy 82: 67–76

    Article  Google Scholar 

  • Hájek, A. (2010). Staying regular. Unpublished manuscript.

  • Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitrechnung. (Ergebnisse Der Mathematik.) Foundations of probability. (2nd ed.) (N. Morrison, Trans., 1956). USA: Chelsea Publishing Company.

  • Komjáth P., Totik V. (2008) Ultrafilters. American Mathematical Monthly 115: 33–44

    Google Scholar 

  • Lauwers, L. (2010). Purely finitely additive measures are non-constructible objects. Working paper, unpublished. Retrieved: May 29, 2010, from http://www.econ.kuleuven.be/ces/discussionpapers/Dps10/DPS1010.pdf.

  • Lavine S. (1995) Finite mathematics. Synthese 103: 389–420

    Article  Google Scholar 

  • 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, California

    Google Scholar 

  • Lewis D. (1986) Philosophical papers. Oxford University Press, Oxford

    Google Scholar 

  • Mancosu P. (2009) Measuring the size of infinite collections of natural numbers. The Review of Symbolic Logic 2: 612–646

    Article  Google Scholar 

  • McCall S., Armstrong D. M. (1989) God’s lottery. Analysis 49: 223–224

    Google Scholar 

  • Ramsey F. P. (1926) Truth and probability. In: Braithwaite R. B. (eds) Foundations of mathematics and other essays. Routledge & P. Kegan, London, pp 156–198

    Google Scholar 

  • Robinson A. (1966) Non-standard analysis. North-Holland Publishing Company, New York

    Google Scholar 

  • Rubio J. E. (1994) Optimization and nonstandard analysis. Marcel Dekker, New York

    Google Scholar 

  • Schurz G., Leitgeb H. (2008) Finitistic and frequentistic approximation of probability measures with or without σ-additivity. Studia Logica 89: 257–283

    Article  Google Scholar 

  • Skyrms B. (1980) Causal Necessity. Yale University Press, New Haven

    Google Scholar 

  • Tenenbaum G. (1995) Introduction to analytic and probabilistic number theory. Cambridge University Press, Cambridge

    Google Scholar 

  • Truss J. (1997) Foundations of mathematical analysis. Clarendon Press, Oxford

    Google Scholar 

  • Väth M. (2007) Nonstandard analysis. Springer, Berlin

    Google Scholar 

  • Williamson T. (2007) How probable is an infinite sequence of heads?. Analysis 67: 173–180

    Google Scholar 

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Acknowledgements

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).

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Correspondence to Sylvia Wenmackers.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Wenmackers, S., Horsten, L. Fair infinite lotteries. Synthese 190, 37–61 (2013). https://doi.org/10.1007/s11229-010-9836-x

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  • DOI: https://doi.org/10.1007/s11229-010-9836-x

Keywords

  • Foundations of probability
  • Non-standard analysis
  • Countable additivity
  • Infinity