My question is about Richard von Mises' frequentist approach to probability, but interpreted through Karl Popper's words. Excerpt from the latter's book, "The Logic of Scientific Discovery", section 51 (emphasis mine):
Objections have been raised against combining the axiom of convergence with the axiom of randomness on the ground that it is inadmissible to apply the mathematical concept of a limit, or of convergence, to a sequence which by definition (that is, because of the axiom of randomness), must not be subject to any mathematical rule or law. For the mathematical limit is nothing but a characteristic property of the mathematical rule or law by which the sequence is determined. It is merely a property of this rule or law if, for any chosen fraction arbitrarily close to zero, there is an element in the sequence such that all elements following it deviate by less than that fraction from some definite value - which is then called your limit.
It is the last part where I got a bit confused.
If we take the following sequence:
0 1 1 0 0 0 1 1 1 0 1 0 1 0
We get the following sequence of relative frequencies correlated with the appearance of "1" in the sequence:
0 $\frac{1}{2}$ $\frac{2}{3}$ $\frac{2}{4}$ $\frac{2}{5}$ $\frac{2}{6}$ $\frac{3}{7}$ $\frac{4}{8}$ $\frac{5}{9}$ $\frac{5}{10}$ $\frac{6}{11}$ $\frac{6}{12}$ $\frac{7}{13}$ $\frac{7}{14}$
The limit for the example above seems to be $\frac{1}{2}$ - that's where the sequence is converging towards.
What did Popper mean by "arbitrarily choosing a fraction" and "deviate", and how can one apply his argument to the frequency-sequence above?