. It follows that
divided by
may have any value from 1 up to
.
From the ambiguity of subtraction and division it results that negative numbers and ratios cannot be extended to infinite numbers. Addition, multiplication, and exponentiation proceed quite satisfactorily, but the inverse operations—subtraction, division, and extraction of roots—are ambiguous, and the notions that depend upon them fail when infinite numbers are concerned.
The characteristic by which we defined finitude was mathematical induction, i.e. we defined a number as finite when it obeys mathematical induction starting from 0, and a class as finite when its number is finite. This definition yields the sort of result that a definition ought to yield, namely, that the finite numbers are those that occur in the ordinary number-series 0, 1, 2, 3, ... But in the present chapter, the infinite numbers we have discussed have not merely been non-inductive: they have also been reflexive. Cantor used reflexiveness as the definition of the infinite, and believes that it is equivalent to non-inductiveness; that is to say, he believes that every class and every cardinal is either inductive or reflexive. This may be true, and may very possibly be capable of proof; but the proofs hitherto offered by Cantor and others (including the present author in former days) are fallacious, for reasons which will be explained when we come to consider the "multiplicative axiom." At present, it is not known whether there are classes and cardinals which are neither reflexive nor inductive. If
