Assuming that the number of individuals in the world is , the number of classes of individuals will be . This is in virtue of the general proposition mentioned in Chapter VIII. that the number of classes contained in a class which has members is . Now is always greater than . Hence the number of classes in the world is greater than the number of individuals. If, now, we suppose the number of individuals to be 9, as we did just now, the number of classes will be , i.e. 512. Thus if we take our numbers as being applied to the counting of classes instead of to the counting of individuals, our arithmetic will be normal until we reach 512: the first number to be null will be 513. And if we advance to classes of classes we shall do still better: the number of them will be , a number which is so large as to stagger imagination, since it has about 153 digits. And if we advance to classes of classes of classes, we shall obtain a number represented by 2 raised to a power which has about 153 digits; the number of digits in this number will be about three times . In a time of paper shortage it is undesirable to write out this number, and if we want larger ones we can obtain them by travelling further along the logical hierarchy. In this way any assigned inductive cardinal can be made to find its place among numbers which are not null, merely by travelling along the hierarchy for a sufficient distance.[26] [26]On this subject see Principia Mathematica, vol. II. * 120 ff. On the corresponding problems as regards ratio, see ibid., vol. III. * 303 ff. As regards ratios, we have a very similar state of affairs. If a ratio is to have the expected properties, there must be enough objects of whatever sort is being counted to insure that the null-class does not suddenly obtrude itself. But this can be insured, for any given ratio , without the axiom of [Pg 133] infinity, by merely travelling up the hierarchy a sufficient distance. If we cannot succeed by counting individuals, we can try counting classes of individuals; if we still do not succeed, we can try classes of classes, and so on. Ultimately, however few individuals there may be in the world, we shall reach a stage where there are many more than objects, whatever inductive number may be. Even if there were no individuals at all, this would still be true, for there would then be one class, namely, the null-class, 2 classes of classes (namely, the null-class of classes and the class whose only member is the null-class of individuals), 4 classes of classes of classes, 16 at the next stage, 65,536 at the next stage, and so on. Thus no such assumption as the axiom of infinity is required in order to reach any given ratio or any given inductive cardinal.