popper-logic-scientific-discovery

appendix *vi 371
which from the very beginning behaves in a ‘reasonably random-like’
fashion.
But this phrase, ‘from the very beginning’, creates its own problem.
Is the sequence 010110 random-like? Clearly, it is too short for us to say
yes or no. But if we say that we need a long sequence for deciding a
question of this kind then, it seems, we unsay what we have said
before: it seems that we retract the phrase ‘from the very beginning’.
(8) The solution of this difficulty is the construction of an ideally
random sequence—one which for each beginning segment, whether short
or long, is as random as the length of the segment permits; or in other
words, a sequence whose degree n of randomness (that is, its nfreedom
from after-effects) grows with the length of the sequence as
quickly as is mathematically possible.
How to construct a sequence of this kind has been shown in appendix
iv of the book. (See especially note *1 to appendix iv, with a
reference to an as yet unpublished paper by Dr. L. R. B. Elton and
myself.)
(9) The infinite set of all sequences answering this description may
be called the ideal type of random alternatives with equal distribution.
(10) Although no more is postulated of these sequences than that
they are ‘strongly random’—in the sense that the finite commencing
segments would pass all tests of randomness—they can easily be shown to
possess frequency limits, in the sense usually demanded by frequency
theories. This solves in a simple manner one of the central problems of
my chapter on probability—elimination of the limit axiom, by way of
a reduction of the limit-like behaviour of the sequences to their
random-like behaviour in finite segments.
(11) The construction may quite easily be extended into both directions
of the one-dimensional case, by correlating the first, second, . . .
of the odd numbered elements with the first, second, . . . place of the
positive direction, and the first, second, . . . of the even numbered
elements with the first, second, . . . place of the negative direction; and
by similar well-known methods, we can extend our construction to the
cells of an n-dimensional space.
(12) While other frequency theorists—especially von Mises, Copeland,
Wald, and Church—were mainly interested in defining random
sequences in the most severe way by excluding ‘all’ gambling systems