There is one advantage in the above direct argument, as against deduction from Zermelo's theorem, that the above argument does not demand the universal truth of the multiplicative axiom, but only its truth as applied to a set of classes. It may happen that the axiom holds for classes, though not for larger numbers of classes. For this reason it is better, when [Pg 129] it is possible, to content ourselves with the more restricted assumption. The assumption made in the above direct argument is that a product of factors is never zero unless one of the factors is zero. We may state this assumption in the form: " is a multipliable number," where a number is defined as "multipliable" when a product of factors is never zero unless one of the factors is zero. We can prove that a finite number is always multipliable, but we cannot prove that any infinite number is so. The multiplicative axiom is equivalent to the assumption that all cardinal numbers are multipliable. But in order to identify the reflexive with the non-inductive, or to deal with the problem of the boots and socks, or to show that any progression of numbers of the second class is of the second class, we only need the very much smaller assumption that is multipliable. It is not improbable that there is much to be discovered in regard to the topics discussed in the present chapter. Cases may be found where propositions which seem to involve the multiplicative axiom can be proved without it. It is conceivable that the multiplicative axiom in its general form may be shown to be false. From this point of view, Zermelo's theorem offers the best hope: the continuum or some still more dense series might be proved to be incapable of having its terms well ordered, which would prove the multiplicative axiom false, in virtue of Zermelo's theorem. But so far, no method of obtaining such results has been discovered, and the subject remains wrapped in obscurity. [Pg 130] CHAPTER XIII THE AXIOM OF INFINITY AND LOGICAL TYPES THE axiom of infinity is an assumption which may be enunciated as follows:— "If be any inductive cardinal number, there is at least one class of individuals having terms."