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

SECTION E.

PRODUCTS AND SUMS OF CLASSES.
Summary of Section E.
In the present section, we make an extension of a n, a vu, R A S,
R u S. Given a class of classes, say Kc, the product of Kc (which is denoted
by p'K) is the common part of all the members of K, i.e. the class consisting of
those terms which belong to every member of K. The definition is
p'Kc= (aE6K. a.xe a) Df.
If K has only two members, a and /3 say, p'K =anr3. If K has three
members, a, /, y, then p'tc = a n /3 n 7y; and so on. But this process can only
be continued to a finite number of terms, whereas the definition of p'K does
not require that Kc should be finite. This notion is chiefly important in connection with tie lower limits of series. For example, let X be the class of
rational numbers whose square is greater than 2, and let " xMy " mean "x < y,
where x and y are rationals." Then if x e X, M'x will be the class of rationals
less than x. Thus M"X will be the class of such classes as M'x, where x e X.
Thus the product of M"x, which we call p'M"x, will be the class of rationals
which are less than every member of X, i.e. the class of rationals whose squares
are less than 2. Each member of M"x is a segment of the series of rationals,
and p'M"X is the lower limit of these segments. It is thus that we prove the
existence of lower limits of series of segments.
Similarly the sum of a class of classes Ke is defined as the class consisting
of all terms belonging to some member of K; i.e.
s'c = {(a). a K.x e a Df,
i.e. x belongs to the sum of K if x belongs to some K. This notion plays the
samne part for upper limits of series of segments as p'/ plays for lower limits.
It has, however, many more other uses than pKc, and is altogether a more important conception. Thus in cardinal arithmetic, if no two members of K have
any term in common, the aritlmetical sum of the numbers of members possessed
by the various members of K is the number of members possessed by s'Kc.