Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*55. ORDINAL COUPLES.

Summary of *55.
Ordinal couples, which are now to be considered, are much more important, even in cardinal arithmetic, than cardinal couples. Their properties
are in part analogous to those of cardinal couples, but in part also to those of
unit classes; for they are the smallest existent relations, just as unit classes
are the smallest existent classes. The properties which are analogous to
those of unit classes do not demand that the two terms of the couple should
be distinct, i.e. they hold for t'x ' t'x as well as for t'x T tiy (where x $ y);
on the other hand, the properties which are analogous to those of cardinal
couples do in general demand that the two terms of the ordinal couple should
be distinct.
The notation i'x tL'y is cumbrous, and does not readily enable us to
exhibit the couple as a descriptive function of x for the argument y, or vice
versa. We therefore introduce a new symbol, "x 4, y," for the couple. In a
couple x, y, we shall call x the referent of the couple, and y the relatum.
In virtue of the definitions in *38, this gives rise to two relations, x J and, y; hence we obtain the notations x, "/3, y, y", a y, a "3 and so on,
which will be much used in the sequel. It should be observed that x l "/
means the class of ordinal couples in which x is referent and a member of 3
is relatum, while I y"a or a y denotes the class of couples having y as
relatum and a member of a as referent; a J, "1 denotes all such classes of
couples as I y"a, where y is any member of /3; and in virtue of *40'7,
s'a "/3 denotes all ordinal couples of which the referent is a member of a,
while the relatum is a member of /. This is a very important class, which
will be used to define the product of two cardinal numbers; for it is evident
that the number of members of s'a 4 "/3 is the product of the number of
members of a and the number of members of /.
The first few propositions of the present number are immediate consequences of the definition of x 4 y and the notations introduced in *38. We
then proceed to various elementary properties of the relation x 4 y, of which
the most used are the following: