We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Bartha P., Hitchcock C.: The shooting-room paradox and conditionalizing on measurably challenged sets. Synthese 118, 403–437 (1999)
V. Benci, I numeri e gli insiemi etichettati, Conferenze del seminario di matematica dell’ Universita’ di Bari, vol. 261, Laterza, Bari, Italy, 1995, p. 29.
V. Benci and M. Di Nasso, Alpha-theory: an elementary axiomatic for nonstandard analysis, Expositiones Mathematicae 21 (2003), 355–386.
Benci V.: Numerosities of labelled sets: a new way of counting. Advances in Mathematics 173, 50–67 (2003)
Benci V., Di Nasso M., Forti M.: An Aristotelian notion of size. Annals of Pure and Applied Logic 143, 43–53 (2006)
V. Benci, The eightfold path to nonstandard analysis, Nonstandard Methods and Applications in Mathematics (N. J. Cutland, M. Di Nasso, and D. A. Ross, eds.), Lecture Notes in Logic, vol. 25, Association for Symbolic Logic, AK Peters, Wellesley, MA, 2006, pp. 3–44.
V. Benci, L. Horsten, and S. Wenmackers, Infinitesimal probabilities, In preparation, 2012.
E. Bottazzi, ω-Theory: Mathematics with Infinite and Infinitesimal Numbers, Master thesis, University of Pavia, Italy, 2012.
Cutland N.: Nonstandard measure theory and its applications. Bulletin of the London Mathematical Society 15, 529–589 (1983)
B. de Finetti, Theory of probability, Wiley, London, UK, 1974, Translated by: A. Machí and A. Smith.
T. Gilbert and N. Rouche, Y a-t-il vraiment autant de nombres pairs que de naturels?, Méthodes et Analyse Non Standard (A. Pétry, ed.), Cahiers du Centre de Logique, vol. 9, Bruylant-Academia, Louvain-la-Neuve, Belgium, 1996, pp. 99–139.
Hájek A.: What conditional probability could not be. Synthese 137, 273–323 (2003)
J.B. Kadane, M.J. Schervish, and T. Seidenfeld, Statistical implications of finitely additive probability, Bayesian Inference and Decision Techniques (P.K. Goel and A. Zellnder, eds.), Elsevier, Amsterdam, The Netherlands, 1986.
H.J. Keisler, Foundations of infinitesimal calculus, University of Wisconsin, Madison, WI, 2011.
H.J. Keisler and S. Fajardo, Model theory of stochastic processes, Lecture Notes in Logic, Association for Symbolic Logic, 2002.
A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse der Mathematik, 1933, Translated by N. Morrison, Foundations of probability. Chelsea Publishing Company, 1956 (2nd ed.).
Levi I.: Coherence, regularity and conditional probability. Theory and Decision 9, 1–15 (1978)
D. K. Lewis, A subjectivist’s guide to objective chance, Studies in Inductive Logic and Probability (R. C. Jeffrey, ed.), University of California Press, Berkeley, CA, 1980, pp. 263–293.
P.A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society 211 (1975), 113–122.
E. Nelson, Radically elementary probability theory, Princeton University Press, Princeton, NJ, 1987.
B. Skyrms, Causal necessity, Yale University Press, New Haven, CT, 1980.
R. Weintraub, How probable is an infinite sequence of heads? A reply to Williamson, Analysis 68 (2008), 247–250.
S. Wenmackers and L. Horsten, Fair infinite lotteries, Accepted in Synthese, DOI: 10.1007/s11229-010-9836-x, 2010.
Williamson T.: How probable is an infinite sequence of heads?. Analysis 67, 173–180 (2007)
About this article
Cite this article
Benci, V., Horsten, L. & Wenmackers, S. Non-Archimedean Probability. Milan J. Math. 81, 121–151 (2013). https://doi.org/10.1007/s00032-012-0191-x
Mathematics Subject Classification (2010)
- axioms of Kolmogorov
- nonstandard models
- fair lottery
- non-Archimedean fields