Probability in the alternative set theory. (English) Zbl 0677.03039

Kalina, Martin

Commentat. Math. Univ. Carol. 30, No. 2, 347-356 (1989).

The author uses results from his preceding paper: “A sequential approach to the construction of measures” [ibid. 30, No.1, 121-128 (1989)] for the introduction of statistical notions (from random variable to stochastic process) in the framework of alternative set theory (AST). In the case that the “cardinality” of the domain of a random variable is a nonadditive cut (this corresponds to the case of the positive measure in the Loeb measure construction), all the introduced notions behave analogously as those, e.g., described in a paper of R. M. Anderson [Isr. J. Math. 25, 15-46 (1976; Zbl 0353.60052)]. But the author’s main interest lies in the case that the “cardinality” is an additive cut. Then the Loeb measure approach needs the choice of so large basic \({}^*finite\) set (in which the considered external set - e.g. the domain of a random variable - is embedded) that the measure of the considered external set is 0. The value (a real number) of the considered quantity is then undetermined and the author worked out [loc. cit.] a method how to determine the value of the measure depending on the sequence of \({}^*finite\) internal sets approximating the considered external set. The author understands this approximating sequence as the way of measurement. Statistical notions are developed on this basis, and surprising properties are presented. The author expresses his hope that this approach can be used for modeling some phenomena from the theory of elementary particles, but he does not yield a sufficiently developed mathematical apparatus for this. There are some misprints in the paper decreasing its readability.

Reviewer: K.Čuda

MSC:

03E70	Nonclassical and second-order set theories
60A10	Probabilistic measure theory
60G05	Foundations of stochastic processes
60A05	Axioms; other general questions in probability
03H10	Other applications of nonstandard models (economics, physics, etc.)

Keywords:

probability distribution; interval indiscernibility equivalence; random variable; stochastic process; alternative set theory

Citations:

Zbl 0353.60052