popper-logic-scientific-discovery

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new appendices
in the widest possible sense of the word ‘all’ (that is, in the widest
sense compatible with a proof that random sequences so defined exist),
my aim has been quite different. I wished from the beginning to
answer the objection that randomness is compatible with any finite commencing
segment; and I wished to describe sequences that arise from
random-like finite sequences, by a transition to infinity. I hoped by this
method to achieve two things: to keep close to that type of sequence
which would pass statistical tests of randomness, and to prove the limit
theorem. Both have been done now, as here indicated under point (8),
with the help of the construction given in my old appendix iv. But I
have meanwhile found that the ‘measure-theoretical approach’ to
probability is preferable to the frequency interpretation (see my Postscript,
chapter *iii), both for mathematical and philosophical reasons.
(The decisive point is connected with the propensity interpretation of
probability, fully discussed in my Postscript.) I therefore do not think any
longer that the elimination of the limit axiom from the frequency
theory is very important. Still, it can be done: we can build up the
frequency theory with the help of the ideal type of the random
sequences constructed in appendix iv; and we can say that an empirical
sequence is random to the extent to which tests show its statistical
similarity to an ideal sequence.
The sequences admitted by von Mises, Copeland, Wald, and Church
are not necessarily of this kind, as mentioned above. But it is a fact that
any sequence ever rejected on the basis of statistical tests for being not
random may later turn into an admissible random sequence in the
sense of these authors.
Addendum, 1967
(13) Today, some years after reaching a solution of this old problem
which would have satisfied me in 1934, I no longer believe in the
importance of the fact that a frequency theory can be constructed
which is free from all the old difficulties. Yet I still think it important
that it is possible to characterize randomness as a type of order, and that
we can construct models of random sequences.
(14) It is significant that ideally random sequences, as described here
under (8) to (10), satisfy the formal system of appendices *iv and *v,