6.031 The theory of classes is completely superfluous in mathematics.

That the generality that we need in mathematics is not the accidental [contingent] kind hangs together with this.

Russell says, “the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition secures definiteness and indubitableness.” (Introduction to Mathematical Philosophy, p. 18) But then the number 2 depends on the existence of a class of couples, i.e. on the existence of couples. And the number 1000 depends on the existence of 1000 things. And so on. But Russell admits that “Logical propositions are such as can be known a priori, without study of the actual world,” and it is not logically necessary that even one thing exists, he says, let alone 1000, or, even worse, infinity. (ibid., p. 204)