On the concept of a random sequence
Alonzo Church · Bulletin of the American Mathematical Society · 1940
Von Mises has based his frequency theory of probability on the notion of a Kollektiv, 2 that is, of an infinite sequence of trials of an event whose possible outcomes have each a definite probability but otherwise appear entirely at random.(Convenient illustrative examples are an infinite sequence of tosses of a coin, an infinite sequence of rolls of a die, and the like.) 3 Abstractly the Kollektiv may be represented by an infinite sequence of points of an appropriate space, the Merkmalraum.Or if the number of possible outcomes of a trial is finite (and it may well be argued that this is always the case for any actual physical observation 4 ), it is sufficient to employ an infinite sequence of natural numbers which are less than a fixed natural number.This infinite sequence-of points or of natural numbers-satisfies certain conditions which correspond to those appearing in the description of a Kollektiv as just given, and which we shall express by saying that it is a random sequence (regellose Folge).For the present purpose it is largely sufficient to confine attention to the case that each trial has only two possible outcomes, as with the toss of a coin adjudged as falling heads or tails, or the roll of a die adjudged as showing or not showing an ace.The Kollektiv may then be represented abstractly by a random sequence of O's and l's: in the case of the coin, for instance, we may let 1 correspond to the fall of heads and 0 to tails.The definition of a random sequence of O's and 1's as given by von Mises may perhaps be put in the following form:1 Presented to the Society, April 8, 1939. 2 Richard von Mises, Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, vol. 5 (1919), pp.52-99; Wahrscheinlichkeit, Statistik und Wahrheit, Vienna, 1928; Wahrscheinlichkeitsrechnung, Leipzig and Vienna, 1931; and see especially the second edition of Wahrscheinlichkeit, Statistik und Wahrheit, Vienna, 1936, for its account of the objections which have been raised to von Mises's theory and the alternatives which have been proposed.