The Principles of Mathematics (1903)

§ 72

We agreed in the preceding chapter that there are not different ways of denoting, but only different kinds of denoting concepts and correspondingly different kinds of denoted objects. We have discussed the kind of denoted object which constitutes a class; we have now to consider the kind of denoting concept.(§ 72 ¶ 1)

The consideration of classes which results from denoting concepts is more general than the extensional consideration, and that in two respects. In the first place it allows, what the other practically excludes, the admission of infinite classes; in the second place it introduces the null concept of a class. But, before discussing these matters, there is a purely logical point of some importance to be examined.(§ 72 ¶ 2)

If u be a class-concept, is the concept all u's analyzable into two constituents, all and u, or is it a new concept, defined by a certain relation to u, and no more complex than u itself? We may observe, to begin with, that all u's is synonymous with u's, at least according to a very common use of the plural. Our question is, then, as to the meaning of the plural. The word all has certainly some definite meaning, but it seems highly doubtful whether it means more than the indication of a relation. All men and all numbers have in common the fact that they both have a certain relation to a class-concept, namely to man and number respectively. But it is very difficult to isolate any further element of all-ness which both share, unless we take as this element the mere fact that both are concepts of classes. It would seem, then, that all u's is not validly analyzable into all and u, and that language, in this case as in some others, is a misleading guide. The same remark will apply to every, any, some, a, and the.(§ 72 ¶ 3)

It might perhaps be thought that a class ought to be considered, not merely as a numerical conjunction of terms, but as a numerical conjunction denoted by the concept of a class. This complication, however, would serve no useful purpose, except to preserve Peano's distinction between a single term and the class whose only term it is--a distinction which is easy to grasp when the class is identified with the class-concept, but which is inadmissible in our view of classes. It is the same entity as when not so considered, or else is a complex of denoting together with the object denoted; and the object denoted is plainly what we mean by a class.(§ 72 ¶ 4)

With regard to infinite classes, say the class of numbers, it is to be observed that the concept all numbers, though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity. Throughout the discussions of infinity in later Parts of the present work, this remark should be borne in mind: if it is forgotten, there is an air of magic which causes the results obtained to seem doubtful.(§ 72 ¶ 5)