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

*114. THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES.

Summary of *114.
The kind of multiplication defined in *113 cannot be extended beyond a
finite number of factors. We therefore, as in the case of addition, introduce
another definition, defining the product of the numbers of a class of classes,
and capable of being applied to an infinite number of factors. We define the
product of the numbers of members of c as Nc'ea'c; thus we put
IINc'K = Nc'ea'K Df.
It is to be observed that IINc'K is not a function of Nc"Kc, because, if two
members of K have the same number, this will count only once in Nc"cK, but
will count twice in IINc'c.
It is very easy to see that, in case K is finite, Nc'e4'K will be what
we should ordinarily regard as the product of the numbers of members of Kc.
For suppose (e.g.)
K = t-a u lt/ tu 1y,
where a,13. a y./38 y. Then
E~'c= R x(X, y, z). R = xa a wy, v3 z 4. xEa. y. y].
Thus if R is a member of ea:K, R is determinate when x, y, z are given,
x, y, z being the referents to a,,, y. Whether a, /, y overlap or not, the choice
of any one of x, y,z is entirely independent of the choice of the other two,
and therefore the total number of choices possible is obviously the product of
the numbers of a, 3, y. Thus our definition will not conflict with what
is commonly understood by a product.
The propositions of this number are less numerous and less important
than those of *113. We shall deal first with products of a single factor, and
products in which one factor is null (*114'2 —27). We shall then deal
(*114'3 —36) with the relations between the sort of multiplication here
defined and the sort defined in *113. Then we have a few propositions
(*114*4-'43) showing that unit factors make no difference to the value of a
product. Then we prove (*114-5 —52) that the value of the product is the
same for two classes having double similarity, and then (*114'53-'571) we
give extensions of this result which depend upon the multiplicative axiom.
Finally, we give some new forms of the associative law of multiplication.