A problem of a wholly different nature is connected with the possible purchase by the man with an unlimited income of an enumerable infinity of pairs of boots. If he wished to prove that he had an even number of boots, it would be easy if right boots were distinguishable from left ones, but if the man were a faddist of such a kind that he insisted that his left boots should not be made in any way differently from his right ones, it would not be possible for him to prove the theorem mentioned unless he assumed what is known as "the multiplicative axiom." In fact this axiom shows that it is legitimate to pick out an infinite succession of members of an infinite class in an arbitrary way. In the case of the pairs of boots, each pair contains two members, and if there is no means of distinguishing between them, when we wish to pick out one of them for each of the infinity of pairs, we cannot say which ones we mean to pick out unless we assume, by means of the above axiom, that a particularized member can always be found even with things of each of which it can be said that, like Private James in the _Bab Ballads_, No characteristic trait had he Of any distinctive kind. However, a solution of the puzzle was given by Dr. Dénes König of Budapest. You first prove that there are points in space such that, if P is one of them, not more than a finite number of pairs of boots are such that each centre of mass of the two members of a pair is equidistant from P. Taking a point P of this sort, select from each pair the boot whose centre of mass is nearest P. (There may be a finite number of pairs left over, but they can be dealt with arbitrarily.) Another form of the problem is as follows. Every time the man bought a pair of boots he also bought a pair of socks to go with it; he had an enumerable infinity of pairs of each, and the problem is to prove that he had as many boots as he had socks. In this case the boots, we will suppose, can be divided into right and left, but the socks cannot. Thus there are an enumerable infinity of boots, but the number of the socks cannot be determined without admitting the axiom mentioned above. A further difficulty might arise if the owner of the boots and socks lost one leg in some accident, and told his butler to give away half his socks. Naturally the butler would find great logical difficulties in so doing, and it would seem to be an interesting ethical problem whether he should be dismissed from his situation for failing to prove the multiplicative axiom. Again, if the butler stole a pair of boots, the millionaire would have as many pairs as before, but might have fewer boots. There is as yet no evidence that the number of his boots is equal to or greater than the number of pairs. CHAPTER XXXIII THE RELATIONS OF MAGNITUDE OF CARDINAL NUMBERS