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

SECTION D.

SELECTIONS.
Summary of Section D.
The subject to be considered in this section is important chiefly in
connection with multiplication, both cardinal and ordinal. In order to get
a definition of multiplication which is not confined to the case where the
number of factors is finite, we have to seek a construction by which, from
a given class of classes, K say, we construct another class which, when K is
finite, has that number of terms which, in the usual elementary sense, is
the product of the numbers of terms in the various classes which are members
of K, and which, whether K is finite or not, obeys as many as possible of the
formal laws of multiplication. The usual elementary sense of multiplication
is derived from addition; that is to say,,u x v is to be the number of terms in
scK, where K is a class of, mutually exclusive classes each having v members,
or vice versa. This sense can be extended to any finite number of factors,
but not to an infinite number of factors; hence for a number of factors which
may be infinite we require a different definition, and this is derived from the
theory of selections.
Selections are of two kinds, selections from classes of classes, and selections
from relations. The latter is the more general notion, from which the former
is derived. But as the former is an easier notion, we will begin by explaining
selections from classes of classes.
Given a class of classes Kc, a class F is called a selected class of K when,/ is formed by choosing one term out of each member of K. For example, if
c consists of two members, a and 83, and if x a and y e/, then L'x v t'y is
a selected class of K. If every constituency elects a local man, Parliament
is a selected class of the constituencies. If K is a class of mutually exclusive
classes, i.e. a class no two of whose members have any member in common,
then a selected class consists of only one term from each member of K; i.e. p
is a selected class if,t C s'IC: a E K. ^.lF nr a l.
But if K is not a class of mutually exclusive classes, this does not hold
necessarily; for a term x which is a member of both a and / (where a, / e /)
may be chosen as the representative of a, while some other term may be