Non-well-founded probabilities on streams

Abstract

In the paper we propose non-well-founded probabilities as a kind of fuzzy ones. They are defined on the set of streams. We also show that the set of p-adic numbers can be understood as a set of streams. In the set theory without the axiom of foundation, the powerset is not a Boolean algebra in the general case. Therefore, if we tried to define probabilities on non-well-founded data, i.e. on streams or p-adic numbers, then we couldn't use the Kolmogorovian approach and we should refer to non-Kolmogorovian models of probabilities. Probabilities on streams have a lot of unexpected properties. For instance, p-adic probabilities may be negative rational numbers as well as rational numbers that are larger than 1. Bayes' formula doesn't also hold in the general case for non-well-founded probabilities. © 2008 Springer-Verlag Berlin Heidelberg.