_Zermelo's Assumption_
A famous demonstration by Zermelo rests upon the following assumption: In any aggregate (or the same in each aggregate of an assemblage of aggregates) we can always choose _at random_ an element (even if this assemblage of aggregates should contain an infinity of aggregates). This assumption had been applied a thousand times without being stated, but, once stated, it aroused doubts. Some mathematicians, for instance M. Borel, resolutely reject it; others admire it. Let us see what, according to his last article, Russell thinks of it. He does not speak out, but his reflections are very suggestive.
And first a picturesque example: Suppose we have as many pairs of shoes as there are whole numbers, and so that we can number _the pairs_ from one to infinity, how many shoes shall we have? Will the number of shoes be equal to the number of pairs? Yes, if in each pair the right shoe is distinguishable from the left; it will in fact suffice to give the number 2_n_ - 1 to the right shoe of the _n_th pair, and the number 2_n_ to the left shoe of the _n_th pair. No, if the right shoe is just like the left, because a similar operation would become impossible--unless we admit Zermelo's assumption, since then we could choose _at random_ in each pair the shoe to be regarded as the right.
