popper-logic-scientific-discovery

modified, namely, so as to demand no more than is needed to achieve
this aim. Or to be more precise, my aim is the derivation of the
Binomial Formula (sometimes called ‘Newton’s Formula’), in what I
call its ‘third form’. For from this formula, Bernoulli’s theorem and the
other limit theorems of probability theory can be obtained in the usual
way.
My plan is to work out first a frequency theory for finite classes, and to
develop the theory, within this frame, as far as possible—that is, up to
the derivation of the (‘first’) Binomial Formula. This frequency theory
for finite classes turns out to be a quite elementary part of the theory of
classes. It will be developed merely in order to obtain a basis for
discussing the axiom of randomness.
Next I shall proceed to infinite sequences, i.e. to sequences of events
which can be continued indefinitely, by the old method of introducing
an axiom of convergence, since we need something like it for our
discussion of the axiom of randomness. And after deriving and examining
Bernoulli’s theorem, I shall consider how the axiom of convergence might
be eliminated, and what sort of axiomatic system we should be left with as
the result.
In the course of the mathematical derivation I shall use three different
frequency symbols: F″ is to symbolize relative frequency in finite
classes; F′ is to symbolize the limit of the relative frequencies of an
infinite frequency-sequence; and finally F, is to symbolize objective
probability, i.e. relative frequency in an ‘irregular’ or ‘random’ or
‘chance-like’ sequence.
52 RELATIVE FREQUENCY WITHIN A FINITE CLASS
probability 143
Let us consider a class α of a finite number of occurrences, for example
the class of throws made yesterday with this particular die. This class α,
which is assumed to be non-empty, serves, as it were, as a frame of
reference, and will be called a (finite) reference-class. The number of
elements belonging to α, i.e. its cardinal number, is denoted by ‘N(α)’,
to be read ‘the number of α’. Now let there be another class, β, which
may be finite or not. We will call β our property-class: it may be, for
example, the class of all throws which show a five, or (as we shall say)
which have the property five.