Abstract Models of Probability

Probability theory presents a mathematical formalization of intuitive ideas of independent events and a probability as a measure of randomness. It is based on axioms 1-5 of A.N. Kolmogorov ¹ and their generalizations ². Different formalized refinements were proposed for such notions as events, independence, random value etc., ^2,3, whereas the measure of randomness, i.e. numbers from [0,1], remained unchanged. To be precise we mention some attempts of generalization of the probability theory with negative probabilities ⁴. From another side the physicists tryed to use the negative and even complex values of probability to explain some paradoxes in quantum mechanics ^5,6,7. Only recently, the necessity of formalization of quantum mechanics and their foundations ⁸ led to the construction of p-adic probabilities ^9,10,11, which essentially extended our concept of probability and randomness. Therefore, a natural question arises how to describe algebraic structures whose elements can be used as a measure of randomness. As consequence, a necessity arises to define the types of randomness corresponding to every such algebraic structure. Possibly, this leads to another concept of randomness that has another nature different from combinatorical - metric conception of Kolmogorov. Apparenly, discrepancy of real type of randomness corresponding to some experimental data lead to paradoxes, if we use another model of randomness for data processing ¹². Algebraic structure whose elements can be used to estimate some randomness will be called a probability set Φ. Naturally, the elements of Φ are the probabilities.