Principia mathematica, vol. 1 (of 3)

In this section we begin (*50) by introducing a notation for the relation of identity, as opposed to the function ""; that is, calling the relation of identity I, we put The purpose of this definition is chiefly convenience of notation. The definition enables us to speak of , , , , , etc., which we could not otherwise do.

We next introduce a very important notation, due to Peano, for the class whose only member is . If we took a strictly and purely extensional view of classes, we should naturally suppose this class to be identical with . But in view of the theory of classes explained in *20, it is plain that can never be identical with a class of which it is a member, even when it is the only member of that class. Peano uses the notation "" for the class whose only member is ; we shall alter this to "," following our general notation for descriptive functions. Thus we are to have Hence we take as our definition since this definition gives the desired value of . The properties of are many and important.

It is important to observe that "" means "the only member of ." Thus it exists when, and only when, has one member and no more, in which case is of the form , if x is its only member. Thus "" means the same as "(," and "" means the same as "(." What we call "" is denoted, in Peano's notation, by "."

Classes of the form are called unit classes, and the class of all such classes is called 1. This is the cardinal number 1, according to the definition of cardinal numbers which will be given in *100. The properties of 1, so far as they do not depend upon other cardinals, or upon the fact that 1 is a cardinal, will be studied in *52.

After a number (*53) containing various propositions involving 1 or , we pass to the consideration of cardinal couples (*54) and ordinal couples (*55). A cardinal couple is a class , where . The class of such couples is defined as 2, and will be shown at a later stage (*101) to be a cardinal number. An ordinal couple, which, unlike a cardinal couple, involves an order as between its members, is defined as a relation (cf. *35·04), where we may either add or not. The properties of ordinal couples are in part analogous to those of unit classes, in part to those of cardinal couples. In *56, we define the ordinal number 2 (which we denote by , to distinguish it from the cardinal 2) as the class of all ordinal couples , where . It will be shown at a later stage that this is an ordinal number according to our definition of ordinal numbers (*153 and *251).