We may put the matter in another way. To prove that a class has
terms, it is necessary and sufficient to find some way of arranging its terms in a progression. There is no difficulty in doing this with the boots. The pairs are given as forming an
, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pairs unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible. Of course, in the case of objects in space, like socks, we always can find some principle of selection. For example, take the centres of mass of the socks: there will be points
in space such that, with any pair, the centres of mass of the two socks are not both at exactly the same distance from
; thus we can choose, from each pair, that sock which has its centre of mass nearer to