Infinitesimal Probabilities Based on Grossone

Abstract

In finite probability theory probability-zero events occur all the time. Prominent logicians, probability experts and philosophers of probability, including Carnap, Kemeny, Shimony, Savage, De Finetti, Jeffrey, have successfully argued that a sound probability should be regular, that is, only the impossible event should have zero probability. This intuition is shared by physicists too. Totality is another desideratum which means that every event should be assigned a probability. Regularity and totality are achievable in rigorous mathematical terms even for infinite events via hyper-reals valued probabilities. While the mathematics of these theories is not objectionable, some philosophical arguments purport to show that infinitesimal probabilities are inherently problematic. In this paper we present a simpler and natural construction – based on Sergeyev’s calculus with Grossone (in a formalism inspired by Lolli) enriched with infinitesimals – of a regular, total, finitely additive, uniformly distributed probability on infinite sets of positive integers. These probability spaces – which are inspired by and parallel the construction of classical probability – will be briefly studied. In this framework De Finetti fair lottery has the natural solution and Williamson’s objections against infinitesimal probabilities are mathematically refuted.