popper-logic-scientific-discovery

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new appendices
always fill the bill. That this is not so is shown by an example S 2
containing, in addition to the elements of S 1, the elements (plus the settheoretic
products of any two elements and the set-theoretic complement
of any one element) of the sequence B = b 1, b 2, where b n =
(0, (2 n − 1)/2 n + 2 ]. It will be easily seen that, although each b n satisfies
condition (i) for the product-element of A, none of them satisfies condition
(ii); so that in fact, there is no widest element in S 2 that satisfies the
condition (i) for the product-element of A.
Thus S 2 contains neither the set-theoretic product of A, nor a
product-element in our (Boolean) sense. But S 1, and all the systems
obtained by adding to S 1 a finite number of new intervals (plus products
and complements) will contain a product-element of A in our
sense but not in the set-theoretic sense, unless, indeed, we add to S 1 the
missing half-interval h = (0, 1 2].
Remembering that the emptiness of an element a may be characterized
in our system by p(ā, a) ≠ 0, we can now define an ‘admissible
system S’ and a ‘Borel field of probabilities S’, as follows.
(i) A system S that satisfies our postulates 2 to 4 is called an admissible
system if, and only if, S satisfies our set of postulates and in addition
the following defining condition.
Let bA = a 1b, a 2b, . . . be any decreasing sequence of elements of S.
(We say in this case that A = a 1, a 2, . . . is ‘decreasing relative to b’.) Then
if the product element ab of this sequence is in S, 12 then
lim p(a n, b) = p(a, b)
12 I might have added here ‘and if p(ab, ab) ≠ 0, so that ab is empty’: this would have
approximated my formulation still more closely to Kolmogorov’s. But this condition is
not necessary. I wish to point out here that I have received considerable encouragement
from reading A. Rényi’s most interesting paper ‘On a New Axiomatic Theory of Probability’,
Acta Mathematica Acad. Scient. Hungaricae 6, 1955, pp. 286–335. Although I had
realized for years that Kolmogorov’s system ought to be relativized, and although I had
on several occasions pointed out some of the mathematical advantages of a relativized
system, I only learned from Rényi’s paper how fertile this relativization could be. The
relative systems published by me since 1955 are more general still than Rényi’s system
which, like Kolmogorov’s, is set-theoretical, and non-symmetrical; and it can be easily
seen that these further generalizations may lead to considerable simplifications in the
mathematical treatment.