Various arguments professing to prove the existence of infinite classes are given in the Principles of Mathematics, 339 (p. 357). [Pg 137] 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 (both inclusive) is depends upon the assumption that up to and including no number is equal to its successor, which, as we have seen, will not be always true if the axiom of infinity is false. It must be understood that the equation , which might be true for a finite if exceeded the total number of individuals in the world, is quite different from the same equation as applied to a reflexive number. As applied to a reflexive number, it means that, given a class of terms, this class is "similar" to that obtained by adding another term. But as applied to a number which is too great for the actual world, it merely means that there is no class of individuals, and no class of individuals; it does not mean that, if we mount the hierarchy of types sufficiently far to secure the existence of a class of terms, we shall then find this class "similar" to one of terms, for if is inductive this will not be the case, quite independently of the truth or falsehood of the axiom of infinity.