popper-logic-scientific-discovery

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some structural components of a theory of experience
and only two distinct sequences α 2 of adjoining segments; one, which
will be denoted by (A), contains the first, third, fifth, . . . , segments of
α (2), that is, the pairs of α consisting of the numbers 1,2; 3,4; 5,6; . . .
The other, denoted by (B), contains the second, fourth, sixth, . . . ,
segments of α (2), that is, the pairs of elements of α consisting of the
numbers 2,3; 4,5; 6,7; . . . , etc. Now assume that formula (4a) does not
hold for one of the two sequences, (A) or (B), so that the segment (i.e.
the pair) 0,0 occurs too often in, say, the sequence (A); then in sequence
(B) a complementary deviation must occur; that is, the segment 0,0
will occur not often enough (‘too often’, or ‘not often enough’, as compared
with the binomial formula). But this contradicts the assumed
‘absolute freedom’ of α. For if the pair 0,0 occurs in (A) more often
than in (B), then in sufficiently long segments of α the pair 0,0 must
appear more often at certain characteristic distances apart than at other
distances. The more frequent distances would be those which would
obtain if the 0,0 pairs belonged to one of the two α 2-sequences. The less
frequent distances would be those which would obtain if they
belonged to both α 2-sequences. But this would contradict the assumed
‘absolute freedom’ of α; for according to the second binomial formula,
the ‘absolute freedom’ of α entails that the frequency with which a
particular sequence of the length n occurs in any α (n)-sequence depends
only on the number of ones and zeros occurring in it, and not on their
arrangement in the sequence.* 2
This proves (4a); and since this proof can easily be generalized for
any n, the validity of (4) follows; which completes the first step of the
proof.
Second step. The fact that the α n-sequences are ‘absolutely free’ can be
shown by a very similar argument. Again, we first consider α 2sequences
only; and with respect to these it will only be shown, to start
with, that they are 1-free. Assume that one of the two α 2-sequences, e.g.
the sequence (A), is not 1-free. Then in (A) after at least one of the
segments consisting of two elements (a particular α-pair), say after the
* 2 The following formulation may be intuitively helpful: if the 0,0 pairs are more
frequent in certain characteristic distances than in others, then this fact may be easily
used as the basis of a simple system which would somewhat improve the chances of a
gambler. But gambling systems of this type are incompatible with the ‘absolute freedom’
of the sequence. The same consideration underlies the ‘second step’ of the proof.