The Principles of Mathematics (1903)

§ 128

It seems necessary, however, to make a distinction as regards the use of one. The sense in which every object is one, which is apparently involved in speaking of an object, is, as Frege urges[90], a very shadowy sense, since it is applicable to everything alike. But the sense in which a class may be said to have one member is quite precise. A class u has one member when u is not null, and x and y are u's implies x is identical with y. Here the one-ness is a property of the class, which may therefore be called a unit-class. The x which is its only member may be itself a class of many terms, and this shows that the sense of one involved in one term or a term is not relevant to Arithmetic, for many terms as such may be a single member of a class of classes. One, therefore, is not to be asserted of terms, but of classes having one member in the above-defined sense; i.e. u is one, or better u is a unit means u is not null, and x and y are u's implies x and y are identical. The member of u, in this case, will itself be none or one or many if u is a class of classes; but if u is a class of terms, the member of u will be neither none nor one nor many, but simply a term.(§ 128 ¶ 1)

§ 128 n. 1. Grundlagen der Arithmetik, Breslau, 1884, p. 40.