popper-logic-scientific-discovery

probability 161
example of a collective and in that way to show that collectives exist. This
is because an example of an infinite sequence which is to satisfy certain
conditions can only be given by a mathematical rule. But for a collective
in von Mises’s sense there can be, by definition, no such rule, since
any rule could be used as a gambling system or as a system of selection.
This criticism seems indeed unanswerable if all possible gambling
systems are ruled out.* 2
Against the idea of excluding all gambling systems, another objection
may be raised, however: that it really demands too much. If we are going to
axiomatize a system of statements—in this case the theorems of the
calculus of probability, particularly the special theorem of multiplication
or Bernoulli’s theorem—then the axioms chosen should not only
be sufficient for the derivation of the theorems of the system, but also (if
we can make them so) necessary. Yet the exclusion of all systems of selection
can be shown to be unnecessary for the deduction of Bernoulli’s
theorem and its corollaries. It is quite sufficient to demand the exclusion
of a special class of neighbourhood-selection: it suffices to demand that
the sequence should be insensitive to selections according to arbitrarily
chosen n-tuples of predecessors; that is to say, that it should be n-free from
after-effects for every n, or more briefly, that it should be ‘absolutely free’.
I therefore propose to replace von Mises’s principle of the excluded
gambling system by the less exacting requirement of ‘absolute freedom’,
in the sense of n-freedom for every n, and accordingly to define
chance-like mathematical sequences as those which fulfil this requirement.
The chief advantage of this is that it does not exclude all gambling
systems, so that it is possible to give mathematical rules for
constructing sequences which are ‘absolutely free’ in our sense, and
hence to construct examples. (Cf. section (a) of appendix iv.) Thus
Kamke’s objection, discussed above, is met. For we can now prove that
the concept of chance-like mathematical sequences is not empty, and is
therefore consistent.* 3
* 2 It is, however, answerable if any given denumerable set of gambling systems is to be ruled
out; for then an example of a sequence may be constructed (by a kind of diagonal
method). See section *54 of the Postscript (text after note 5), on A. Wald.
* 3 The reference to appendix iv is of considerable importance here. Also, most of the
objections which have been raised against my theory were answered in the following
paragraph of my text.