kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

totality of the second type (§3.1.5), which is easily reduced to a totality of the fourth type,and a simple totality of the second type.In Chapter 4 I shall return to the considerations that guide us when arithmetizing totalities.Here, it is appropriate to note that difficulties and contradictions appear, because, whenestablishing a certain arithmetizing function F(x), we keep at the same time to intuitivenotions incompatible with it. For example, recognizing that F(x) is continuous, we find itdifficult to imagine that the possibility of a definite proposition x = a is incompatible withour assumption; and that, when admitting that possibility, we ought to make a a point ofdiscontinuity of F(x). But it is hardly needed to say that such contradictions between intuitiveand logical conclusions are rather usual in mathematics, and that they cannot be resolved bysome compromise such as “not any proposition of an infinite totality having probability zerois impossible”. Thus, in the theory of functions, we are not embarrassed by the contradictionbetween our intuitive notion of a curve and the existence of continuous functions lacking aderivative; and it will certainly never occur to anybody to assume, that a continuous functionis absolutely arbitrary, and to consider, at the same time, a tangent at some point of the curvedepicting that function.Having any absolutely arbitrary totality of the fourth type of any cardinal number as atotality of all the points of a segment, and of all of their possible joins, we will alwayspreserve, after its arithmetization, only a countable totality of elementary propositions, andwe will be obliged to consider the other elementary propositions impossible. Indeed, therecannot be more than one elementary proposition with probability higher than 1/2; or morethan two of them having probabilities exceeding 1/3 etc.The choice of the elementary propositions which should be considered possible, is in manycases an unsolvable problem. Indeed, who, for example, will be able to indicate thatcountable totality of the points of a segment, which anybody at all had already indicated orchosen, or will indicate or choose (as, for example, 1/2, or 1/2, or ln2, etc)? Nevertheless, itis obvious that this totality is countable 32 whereas all the other numbers ought to beconsidered impossible, because they never were, and never will be actually realized, and,consequently, cannot be realized. This inability is proper; in accord with the demands ofexperience, practice in most cases compels us to abandon the attempts to arithmetizetotalities of the fourth type, and to replace them by those of the second type, naturallywithout violating the principles of the theory.Here is the usual reasoning: When having two equal {congruent} finite segments, theprobabilities that a definite number is contained in either of them are equal. However, thisconsideration is not quite rigorous. The less is the length of the intervals, the moreconsiderable is the inaccuracy of that assumption which cannot be absolutely admitted sinceit would have led us to an arithmetizing function F(z) = z, incompatible, as I showed above,with the realization of definite equalities x = a. On the contrary, when considering ourarithmetization as only approximate; when assuming that the probabilities are not equal butdiffering one from another less than by some very small but not exactly known number , weshould remember that our arithmetization is relatively the less satisfactory the smaller are thesegments (so that, in particular, the probability of the equality x = a is not always equal tozero). We have thus solved the paradox consisting in that, for the arithmetizing function F(z)exactly equal to z, all the totality of the never realizable 33 (impossible) numbers withmeasure 1 would have probability 1 (equal to certainty).When arithmetizing a totality of the fourth type, the issue of determining the probabilitiesof the so-called non-measurable totalities of points should also be indicated. For us, thisproblem does not present difficulties, because, after choosing the arithmetizing function, –that is, after selecting the countable totality of elementary propositions, – any totality ofpoints, whether measurable or not, acquires a probability on the strength of the generalized