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

*115. MULTIPLICATIVE CLASSES AND ARITHMETICAL CLASSES.

Summary of *115.
Whenever c is a class of mutually exclusive classes, e'/c is similar to
D"'eA'c; hence
HNc',c = Nc'D"eC'K.
Now D"eA'K is of the same type as K; and when K is a class of mutually
exclusive classes, D"e'K consists of all classes formed by selecting one
representative from each member of K. It often happens that D"ea'/ is
easier to deal with than ea'K; hence when possible (i.e. when K e Cls2 excl), it
is convenient to use D"e,'K, rather than e4'c, as the standard member of
lINc'K. We therefore put
Prod'K= D"e&'c Df.
We shall call Prod'K the "multiplicative class" of K.
The associative law,
Prod's'Kc sm Prod'Prod"'K,
requires not merely K e Cls2 excl, but also S'K e Cls2 excl. The combination of
these two hypotheses gives a completely disjointed class of classes of classes,
i.e. a class of classes of classes K which can be obtained by dividing a given
class (s's'c) into mutually exclusive portions, and then dividing each of those
portions into mutually exclusive portions. For example, take a square (a class
of points) and divide it by horizontal lines, and then divide each of the resulting rectangles by vertical lines; then the resulting rows of little rectangles
form such a class, each row of rectangles being one member of the class. Such
a class we call an "arithmetical" class, and denote by " Cls3 arithm."
The present number is concerned with the properties of multiplicative
classes and arithmetical classes. Some of these properties will be useful
in dealing with exponentiation.
The present number begins with various propositions concerning Prod'K
which are merely repetitions of previous propositions of *83, *84, *85 or
*113. Thus we have
*115-141. F: g! Prod'K.. s'Prod'K = sK by *83'66,
115-142. F. Prod''a = t"a by *83-7,
*115143. F. Prod't"a = t'a by *83-71,
*115-16.. K e C1s2 excl. ). Prod'K C Nc'K by *100-64,
and various other properties.