kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

addition theorem depending on the points which are included in it and are corresponding tothe elementary propositions.As to the totalities of the second type considered above, they include, because of their verystructure, only such joins 34 , that are reducible to finite or countable joins, so that we do nothave to mention non-measurable totalities; consequently, all the propositions of the totalities,both of the fourth and the second type, acquire quite definite probabilities once thearithmetizing function F(z) is chosen.3.2.7. Arithmetization of Totalities of the Third TypeOn the basis of what was said about totalities of the fourth type, we already know that afterarithmetization only a countable totality of possible elementary propositions can be left in theconsidered totality. The difference between arithmetized totalities of the third and the fourthtypes only consists in that there exist infinite joins making sense in the fourth, but not in thethird type; we do not therefore mention here probabilities of such joins. All the other joinswill have the same probabilities in both totalities.Summarizing all that was said about the arithmetization of infinite totalities, we see that, towhichever type they belong, this procedure is entirely determined by the function F(x) 35 , onwhose choice the very type of the totality also depends because a point of discontinuity ofF(z) corresponds to each elementary proposition, and vice versa. If we admit the generalizedconstructive principle, we obtain, depending on the nature of F(z), totalities of the second andthe fourth types. If, however, we hesitate to attach sense to some infinite joins (andcombinations), our totalities should be attributed to the first or the third type.3.2.8. Arithmetizaton of the Totality of IntegersIntegers and their finite joins provide an example of a totality of the third type. If we linkto them all the possible infinite joins, we obtain a totality of the fourth type with a countabletotality of elementary propositions. Its arithmetization is usually achieved on the basis of theassumption that all numbers are equally possible. This premise however is obviouslyinadmissible because it would have implied that the probability of each number is zero, i.e.,that no number could have been realized, and, in addition, the generalized addition theoremwould have been violated because the sum of the probabilities of a countable totality ofpropositions with probability 0 would have been unity.The difficulty of selecting a law of probability for the numbers depending, in eachconcrete case, on the statement of the problem at hand, cannot justify the choice of a law,even if it is simple, contradicting the main principles of the theory of probability. We mayconsider such a limit of probabilities of some propositions, that corresponds to a graduallyincreasing restricted totality of numbers, under the assumption that in such totalities thenumbers are equally possible, but that limit is not the probability of a definite proposition ofour infinite totality.Another inadmissible assumption, connected with the one just mentioned, is made as oftenas that latter, viz., that the probability for the number N, when divided by a prime number ato provide a remainder , does not depend on the remainder left after dividing N by a primenumber b. Indeed, let o = 0, 1 = 1 be the two possible remainders after dividing N by two; o = 0, 1 = 1 and 2 = 2, the remainders after dividing it by three; etc. Then, according to theassumption, the probabilities of all the infinite combinations ( and and …) are equal, butmost of these combinations are impossible, because, when dividing N by a greater number,all the remainders obtained become equal to N, so that, for example, the combination (0, 1, 0,1, …) of the remainders is impossible. And it would follow that also impossible are thosecombinations which correspond to integers.It is also possible to attach another meaning to all the combinations if we connect each ofthem with the series