Another illustration may help to make the point clearer. We know that . Hence we might suppose that the sum of pairs must have terms. But this, though we can prove that it is sometimes the case, cannot be proved to happen always [Pg 125] unless we assume the multiplicative axiom. This is illustrated by the millionaire who bought a pair of socks whenever he bought a pair of boots, and never at any other time, and who had such a passion for buying both that at last he had pairs of boots and pairs of socks. The problem is: How many boots had he, and how many socks? One would naturally suppose that he had twice as many boots and twice as many socks as he had pairs of each, and that therefore he had of each, since that number is not increased by doubling. But this is an instance of the difficulty, already noted, of connecting the sum of classes each having terms with . Sometimes this can be done, sometimes it cannot. In our case it can be done with the boots, but not with the socks, except by some very artificial device. The reason for the difference is this: Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair. Hence the problem. 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 [Pg 126] 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 . But there is no theoretical reason why a method of selection such as this should always be possible, and the case of the socks, with a little goodwill on the part of the reader, may serve to show how a selection might be impossible.