popper-logic-scientific-discovery

176
some structural components of a theory of experience
64 ELIMINATION OF THE AXIOM OF CONVERGENCE.
SOLUTION OF THE ‘FUNDAMENTAL PROBLEM
OF THE THEORY OF CHANCE’
So far frequency limits have had no other function in our reconstruction
of the theory of probability than that of providing an unambiguous
concept of relative frequency applicable to infinite sequences, so
that with its help we may define the concept of ‘absolute freedom’
(from after-effects). For it is a relative frequency which is required to be
insensitive to selection according to predecessors.
Earlier we restricted our inquiry to alternatives with frequency
limits, thus tacitly introducing an axiom of convergence. Now, so as to
free us from this axiom, I shall remove the restriction without
replacing it by any other. This means that we shall have to construct a
frequency concept which can take over the function of the discarded
frequency limit, and which may be applied to all infinite reference
sequences.* 1
One frequency concept fulfilling these conditions is the concept of a
point of accumulation of the sequence of relative frequencies. (A value a is said to be
a point of accumulation of a sequence if after any given element there
are elements deviating from a by less than a given amount, however
small.) That this concept is applicable without restriction to all infinite
reference sequences may be seen from the fact that for every infinite
alternative at least one such point of accumulation must exist for the
sequence of relative frequencies which corresponds to it. Since relative
frequencies can never be greater than 1 nor less than 0, a sequence of
them must be bounded by 1 and 0. And as an infinite bounded
sequence, it must (according to a famous theorem of Bolzano and
Weierstrass) have at least one point of accumulation. 1
For brevity, every point of accumulation of the sequence of relative
frequencies corresponding to an alternative α will be called ‘a middle
frequency of α’. We can then say: If a sequence α has one and only one
middle frequency, then this is at the same time its frequency limit; and
* 1 In order not to postulate convergence, I appealed in the following paragraph to what
can be demonstrated—the existence of points of accumulation. All this becomes unnecessary
if we adopt the method described in note *1 to section 57, and in appendix *vi.
1 A fact which, surprisingly enough, has not hitherto been utilized in probability theory.