kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

Bernstein’s standpoint regarding infinitely many trials (his §3.2.1) and even his opinionconcerning general mathematical constructions such as convergence almost everywhere.Bernstein’s contribution translated below is hardly known outside Russia. EvenHochkirchen (1999) only mentioned it in his list of Quellen und Fachliteratur but not at all inhis main text.Bernstein had not systematized his memoir. The numbering of the formulas was notthought out, theorems followed one another without being numbered consecutively; notationwas sometimes violated and misprints were left unnoticed. Finally, in §3.2.5 Bernsteinintroduced function F(z) which appeared earlier as F(x). And, what happens time and timeagain in the works of many authors, he had not supplied the appropriate page number in hisepigraph above. I have not been able to correct sufficiently these shortcomings but at least Imethodically numbered the axioms, theorems and corollaries although several propositionsnot called either theorems or corollaries; again, yet others were named principles.* * *The calculation of probabilities is based on several axioms and definitions. Usually,however, these main axioms are not stated sufficiently clearly; it remains therefore an openquestion which assumptions are necessary, and whether they do not contradict one another.The definition itself of mathematical probability implicitly contains a premise (Laplace 1814,p. 4) in essence tantamount to the addition theorem which some authors (Bohlmann ca. 1905,p. 497) assume as an axiom. Consequently, I consider it of some use to explicate here myattempt to justify axiomatically the theory of probability. I shall adhere to a purelymathematical point of view that only demands a rigorous and exhausting statement ofindependent rules not contradicting each other, on whose foundation all the conclusions ofthe theory, regarded as an abstract mathematical discipline, ought to be constructed. It is ofcourse our desire for cognizing the external world as precisely as possible that dictates usthese rules. However, so as not to disturb the strictly logical exposition, I prefer to touch theissue of the philosophical and practical importance of the principles of probability theoryonly in a special supplement at the end of this paper.Chapter 1. Finite Totalities of Propositions1.1. Preliminary Definitions and Axioms1.1.1. Equivalent and Non-Equivalent PropositionsLet us consider a finite or infinite totality of symbols A, B, C, etc which I shall callpropositions. I shall write M = N (N = M) and call M and N equivalent after agreeing that,when performing all the operations defined below on our symbols, it is always possible toreplace M by N and vice versa. In particular, if M = N and M = L, then N = L.Suppose that not all of the given propositions are equivalent, that there exist two such Aand B that A ' B. If the number of non-equivalent propositions is finite, I shall call theirgiven totality finite; otherwise, infinite. In this chapter, I consider only finite totalities.1.1.2. Axioms Describing the Operation (of Partition) Expressed by the Sign “Or”1.1. The constructive principle: If (in the given totality) there exist propositions A and B,then proposition C = (A or B) also exists.1.2. The commutative principle: (A or B) = (B or A ).1.3. The associative principle: [A or (B or C)] = [(A or B) or C] = (A or Bor C).1.4. The principle of tautology: (A or A) = A.By applying the first three principles it is possible to state that, in general, there exists aquite definite proposition H = (A or B or … E). I shall call it a join of propositions A, B, …,E. Each of these is called a particular case of H.