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some structural components of a theory of experience
points out, a gambling system can always be regarded as a systematic
selection.)
Most of the criticism which has been levelled against this axiom
concentrates on a relatively unimportant and superficial aspect of its
formulation. it is connected with the fact that, among the possible
selections, there will be the selection, say, of those throws which come
up five; and within this selection, obviously, the frequency of the fives
will be quite different from what it is in the original sequence. This is
why von Mises in his formulation of the axiom of randomness speaks
of what he calls ‘selections’ or ‘choices’ which are ‘independent of the
result’ of the throw in question, and are thus defined without making
use of the property of the element to be selected. 1 But the many attacks
levelled against this formulation 2 can all be answered merely by pointing
out that we can formulate von Mises’s axiom of randomness without
using the questionable expressions at all. 3 For we may put it, for
example, as follows: The limits of the frequencies in a collective shall
be insensitive both to ordinal and to neighbourhood selection, and also
to all combinations of these two methods of selection that can be used
as gambling systems.* 1
With this formulation the above mentioned difficulties disappear.
Others however remain. Thus it might be impossible to prove that the
concept of a collective, defined by means of so strong an axiom of
randomness, is not self-contradictory; or in other words, that the class
of ‘collectives’ is not empty. (The necessity for proving this has been
stressed by Kamke. 4 ) At least it seems to be impossible to construct an
1 Cf. for example von Mises’s Wahrscheinlichkeit, Statistik und Wahrheit, 1928, p. 25; English
translation, 1939, p. 33.
2 Cf. for instance, Feigl, Erkenntnis 1, 1930, p. 256, where that formulation is described as
‘not mathematically expressible’. Reichenbach’s criticism, in Mathematische Zeitschrift 34,
1932, p. 594 f., is very similar.
3 Dörge has made a similar remark, but he did not explain it.
* 1 The last seven words (which are essential) were not in the German text.
4 Cf. for instance, Kamke, Einführung in die Wahrscheinlichkeitstheorie, 1932, p. 147, and Jahresbericht
der Deutschen mathem. Vereinigung 42, 1932. Kamke’s objection must also be raised
against Reichenbach’s attempt to improve the axiom of randomness by introducing
normal sequences, since he did not succeed in proving that this concept is non-empty. Cf.
Reichenbach, Axiomatik der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 34, 1932,
p. 606.