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 [Pg 87] 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 were such a cardinal, we should not have , but would not be one of the "natural numbers," and would be lacking in some of the inductive properties. All known infinite classes and cardinals are reflexive; but for the present it is well to preserve an open mind as to whether there are instances, hitherto unknown, of classes and cardinals which are neither reflexive nor inductive. Meanwhile, we adopt the following definitions:— A finite class or cardinal is one which is inductive. An infinite class or cardinal is one which is not inductive. All reflexive classes and cardinals are infinite; but it is not known at present whether all infinite classes and cardinals are reflexive. We shall return to this subject in Chapter XII. [Pg 88] CHAPTER IX INFINITE SERIES AND ORDINALS AN "infinite series" may be defined as a series of which the field is an infinite class. We have already had occasion to consider one kind of infinite series, namely, progressions. In this chapter we shall consider the subject more generally. The most noteworthy characteristic of an infinite series is that its serial number can be altered by merely re-arranging its terms. In this respect there is a certain oppositeness between cardinal and serial numbers. It is possible to keep the cardinal number of a reflexive class unchanged in spite of adding terms to it; on the other hand, it is possible to change the serial number of a series without adding or taking away any terms, by mere re-arrangement. At the same time, in the case of any infinite series it is also possible, as with cardinals, to add terms without altering the serial number: everything depends upon the way in which they are added.