, 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 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
