On removal of one undetected paradox in foundations of probability theory

Abstract

The paper points out a logical contradiction that arises in the modern approach to teaching the foundations of the probability theory. This contradiction is of the following form: \U0001d74e = {\U0001d74e}, where \U0001d74e is an arbitrary elementary outcome. In modern mathematics the formation of expressions of the form \U0001d499 = {\U0001d499} is unacceptable. Here {\U0001d499} is a unit set, i.e., a set whose sole element is some element \U0001d499 of an arbitrary nature. This paradox \U0001d74e = {\U0001d74e} is a consequence of the fact that random events and between them also elementary outcomes \U0001d74e are considered as some subsets in the set of all elementary outcomes. It is shown how this contradiction can be easily eliminated by applying the axiomatic method. It is based on the introduction of two simple axioms that impose restrictions on elementary outcomes and on the class of random events under consideration. In addition from these two axioms on the basis of strictly logical reasoning the authors derived the representation of an arbitrary random event in the form of a sum of elementary outcomes favorable to it. Moreover, another non-axiomatic approach to the elimination of this paradox is proposed. In the final part of the paper other variants of the second of the introduced axioms are considered as well.