kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

theoretically certain and its non-occurrence can take place only because of an incompleteaccordance between the actual conditions and our theoretical pattern. Thus, when forming adefinite infinite binary fraction, for example, 8/15 = 0.10001 …, where the limit of therelative number of the units is 1/4, we must state that its composition is incompatible with theassumption that the occurrence of 1 and 0 on each place is equally probable.And it is generally impossible to indicate a method of forming an infinite binary fraction,where the sequence of unities and zeros would have obeyed an infinite number of conditionsimplied by the laws of large numbers. Infinite series composed absolutely arbitrarily,randomly (so that each number is arbitrary in itself) essentially differ from series compiled inaccord with a definite mathematical law no matter how arbitrary it is. A confusion of thesetwo notions, caused by the fact that such differentiation does not exist between random andregular finite series, is one of the main sources of paradoxes to which the theory ofprobability of infinite totalities is leading.Notes1. (§1.1.8). Conversely, the principle of uniqueness follows if we suppose that (A or A ) =; that is, when assuming that a proposition and its negation are solely possible. Indeed, if is compatible with any proposition (excepting O), then α = O, hence ( or O) = and =.2. (§1.1.9). When applying it to the given equation (4), we find that conditions{(A or a or b) or [ A and ( a′or b′ )]} = ,{(A) or a or b) or [ A and ( a or b )]} = are necessary and sufficient for its solvability.3. (§1.2.1). It could have been proved that, also conversely, the assumption that a trueproposition exists, leads to the constructive principle. Thus, for a finite totality {ofpropositions}, this principle and the axiom of the existence of a true proposition areequivalent. {Not axiom but Theorem 1.1.}4. (§1.2.3). For example, the principle of uniqueness is not valid for the system 1, 2, 3, 4,6, 12. The proposition corresponding to the number 2 would have been compatible with allthe propositions and its negation would therefore be only the false proposition. This lastmentionedwould therefore possess the most important property of a true proposition withouthowever being true.5. (§1.3.1). So as not to exclude the false proposition, it is possible to agree that it is a joinof zero elementary propositions, – that it does not contain any of them.6. (§2.1.1). The theory of probability considers only perfect totalities of propositions.7. (§2.1.4). Had we called the fraction m/(n – m) probability, the ratio of the number of theoccurrences of an event to the total number of trials should have been replaced, in theBernoulli theorem for example, by the ratio of the former to the number of the nonoccurrences.The statement of the addition theorem would have also been appropriatelychanged: the probability of (A or B) would have been equal not to the sum of theprobabilities, (p + p 1 ), but to(p + p 1 + 2pp 1 )/(1 – pp 1 ).8. (§2.1.5).{Below, the author specifies his incomplete distinction.}9. (§2.1.5) If p + p 1 > 1, we would have to choose A instead of B and would have thusconvinced ourselves in the inadmissibility of such an assumption.