The Principles of Mathematics (1903)

§ 487

That some modification in that doctrine is necessary, is proved by the argument of KB. p. 444. This argument appears capable of proving that a class, even as one, cannot be identified with the class of which it is the only member. In § 74, I contended that the argument was met by the distinction between the class as one and the class as many, but this contention now appears to me mistaken. For this reason, it is necessary to re-examine the whole doctrine of classes.(§ 487 ¶ 1)

Frege's argument is as follows. If a is a class of more than one term, and if a is identical with the class whose only term is a, then to be a term of a is the same thing as to be a term of the class whose only term is a, whence a is the only term of a. This argument appears to prove not merely that the extensional view of classes is inadequate, but rather that it is wholly inadmissible. For suppose a to be a collection, and suppose that a collection of one term is identical with that one term. Then, if a can be regarded as one collection, the above argument proves that a is the only term of a. We cannot escape by saying that ∈ is to be a relation to the class-concept or the concept of the class or the class as many, for if there is any such entity as the class as one, there will be a relation, which we may call ∈, between terms and their classes as one. Thus the above argument leads to the conclusion that either (α) a collection of more than one term is not identical with the collection whose only term it is, or (β) there is no collection as one term at all in the case of a collection of many terms, but the collection is strictly only many. One or other of these must be admitted in virtue of the above argument.(§ 487 ¶ 2)