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.