members. In this way the attempt to escape from the need for the axiom of infinity breaks down. I do not pretend to have explained the doctrine of types, or done more than indicate, in rough outline, why there is need of such a doctrine. I have aimed only at saying just so much as was required in order to show that we cannot prove the existence of infinite numbers and classes by such conjurer's methods as we have been examining. There remain, however, certain other possible methods which must be considered.
Various arguments professing to prove the existence of infinite classes are given in the Principles of Mathematics, § 339 (p. 357). In so far as these arguments assume that, if
is an inductive cardinal,
is not equal to
, they have been already dealt with. There is an argument, suggested by a passage in Plato's Parmenides, to the effect that, if there is such a number as 1, then 1 has being; but 1 is not identical with being, and therefore 1 and being are two, and therefore there is such a number as 2, and 2 together with 1 and being gives a class of three terms, and so on. This argument is fallacious, partly because "being" is not a term having any definite meaning, and still more because, if a definite meaning were invented for it, it would be found that numbers do not have being—they are, in fact, what are called "logical fictions," as we shall see when we come to consider the definition of classes.
The argument that the number of numbers from 0 to