popper-logic-scientific-discovery

294
appendices
after effect) in the following way. We write down an n-free period for
an arbitrarily chosen n. This period will have a finite number of
terms—say n 1. We now write down a new period which is at least
n 1 − 1-free. Let the new period have the length n 2. In this new period, at
least one sequence must occur which is identical with the previously
given period of length n 1; and we rearrange the new period in such a
way that it begins with this sequence (this is always possible, in accordance
with the analysis of section 55). This we call the second period.
We now write down another new period which is at least n 2 − 1-free
and seek in this third period that sequence which is identical with the
second period (after rearrangement), and then so rearrange the third
period that it begins with the second, and so on. In this way we obtain
a sequence whose length increases very quickly and whose commencing
period is the period which was written down first. This
period, in turn, becomes the commencing sequence of the second
period, and so on. By prescribing a particular commencing sequence
together with some further conditions, e.g. that the periods to be written
down must never be longer than necessary (so that they must be
exactly n i − 1-free, and not merely at least n i − 1-free), this method of
construction may be so improved as to become unambiguous and to
define a definite sequence, so that we can calculate for every term of
the sequence whether it is a one or a zero.* 2 We thus have a
(definite) sequence, constructed according to a mathematical rule,
with frequencies whose limits are,
seen that there can be no shorter generating period of a periodic n-free sequence than
one of the length 2 n + 1 .
Proofs of the validity of the rule of construction here given were found by Dr. L.R.B.
Elton and myself. We intend to publish jointly a paper on the subject.
* 2 To take a concrete example of this construction—the construction of a shortest randomlike
sequence, as I now propose to call it—we may start with the period
(0)
01
of the length n0 = 2. (We could say that this period generates a 0-free alternative). Next
we have to construct a period which is n0 − 1-free, that is to say, 1-free. The method of
note *1, above, yields ‘1100’ as generating period of a 1-free alternative; and this has
now to be so re-arranged as to begin with the sequence ‘01’ which I have here called (0).
The result of the arrangement is
(1)
0 1 1 0