popper-logic-scientific-discovery

appendix iv 295
αF′(1) = αF′(0) = 1 2.
By using the procedure employed in the proof of the third form of
the binomial formula (section 60) or of the theorem of Bernoulli
(section 61), it can be shown (with any degree of approximation) for
whatever frequency value we may choose that there exist sequences which are
‘absolutely free’—provided only that we make the assumption (which
we have just proved) that at least one sequence exists which is absolutely
free.
with n1 = 4. We next construct the n1 − 1-free (i.e. 3-free) period determined by the
method of note *1. It is
1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0
We re-arrange this so that it begins with our commencing sequence (1), which yields
(2)
0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 0
Since n2 = 16, we have next to construct, by the method of note *1, a 15-free period
(3) of the length 2 16 = 65,536. Once we have constructed this 15-free period (3), we
must be able to discover where, in this long period, our sequence (2) occurs. We then
re-arrange (3) so that it commences with (2), and proceed to construct (4), of the
length 2 65,536 .
A sequence constructed in this way may be called a ‘shortest random-like sequence’ (i)
because every step of its construction consists in the construction, for some n, of a
shortest n-free period (cf. note *1 above), and (ii) because the sequence is so constructed
that, whatever the stage of its construction, it always begins with a shortest n-free period.
As a consequence, this method of construction ensures that every beginning piece of the
length
is a shortest n-free period for the largest possible n (i.e. for n = (log 2m) − 1).
This property of ‘shortness’ is very important; for we can always obtain n-free, or
absolutely free, sequences with equidistribution which commence with a finite segment
of any chosen length m such that this finite segment has no random character but consists
of say, only zeros, or only ones, or of any other intuitively ‘regular’ arrangement; which
shows that for applications, the demand for n-freedom, or even absolute freedom, is not
enough, and must be replaced by something like a demand for n-freedom, becoming
manifest from the beginning; which is, precisely, what a ‘shortest’ random-like sequence
achieves, in the most radical fashion possible. Thus they alone can set an ideal standard
for randomness. For these ‘shortest’ sequences, convergence can be proved immediately, as
opposed to the examples under (b) and (c) below. See also appendix *vi.