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Postulating the theory of experience and chance as a theory of co~events (co~beings)

Vorobyev, Oleg Yu. (2016): Postulating the theory of experience and chance as a theory of co~events (co~beings). Published in: Proceedings of the XV FAMEMS-2016 Conference and the Workshop on Hilbert's sixth problem, Krasnoyarsk, Russia (30 September 2016): pp. 25-43.

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Abstract

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author’s main idea was the natural inclusion of Kolmogorov’s axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The analogy between the measure of a set and the probability of an event has become clear for a long time. This analogy also allows further evolution: the measure of a set is completely analogous to the believability of an event. In order to postulate the theory of experience and chance on the basis of this analogy, you just need to add to the Kolmogorov probability theory its dual reflection — the believability theory, so that the theory of experience and chance could be postulated as the certainty (believability-probability) theory on the Cartesian product of the probability and believability spaces, and the central concept of the theory is the new notion of co~event as a measurable binary relation on the Cartesian product of sets of elementary incomes and elementary outcomes. Attempts to build the foundations of the theory of experience and chance from this general point of view are unknown to me, and the whole range of ideas presented here has not yet acquired popularity even in a narrow circle of specialists; in addition, there was still no complete system of the postulates of the theory of experience and chance free from unnecessary complications. Postulating the theory of experience and chance can be carried out in different ways, both in the choice of axioms and in the choice of basic concepts and relations. If one tries to achieve the possible simplicity of both the system of axioms and the theory constructed from it, then it is hardly possible to suggest anything other than axiomatization of concepts co~event and its certainty (believability-probability). The main result of this work is the axiom co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work, a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment.

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