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

502

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

Thus they are four in number, i.e. the number of members of eaK is the
product of the number of members of a and the number of members of 3.
A similar process would show that our definition of the product agrees
with the usual definition in any case in which all the numbers concerned
are finite.
Selections from relations are an obvious generalization of selections from
classes of classes. We had above
e^' = (1 - Cls) n Rl'e I'K.
We put, generally,
4 -Pac = (1 - Cls) n RIP n ('t,
which we derive from the definition
4 -PA = Kx = (1 - Cls)n R1P n CK Df.
This is the fundamental definition in the subject of selections. We have, in
virtue of this definition,
F: R e P'K. =. R e 1 - Cls. R G P. (R = K.
When K=(I'P, we may call Pa'K the class of selections from P. Thus
generally, PCK is the class of selections from P c K provided K C ('P; and if
this condition is not fulfilled, PAK = A. We may call the class PA'K the
class of "P-selections from c." The class of "e-selections from K" will be
what we previously called the class of "selective relations of K."
It will be observed that we have
R Pa'K. y e K. D. R'y eP'y.
Thus if P"cK is a class of mutually exclusive classes, D'R selects one from
each of these classes, and is therefore a selective class of P"K; hence in
this case
D"PKC = D"E''P"CK.
In Cardinal Arithmetic, e,'K is the important notion, and the more
general notion PA'K is seldom required. In Ordinal Arithmetic, FA'K is the
important notion. It will be seen that
R e F'K.. R e 1 -- CIs. R C F. ('R = K.
Thus Fa'K is only significant when K is a class of relations; in this case
we have
ReFAi'K. QeC. DK.R'QEC'Q.
Thus R chooses a representative member of the field of every member
of c. The most important case is when K is of the form C'P, where P is
a serial relation whose field consists of serial relations. Then FA'C'P becomes
the field of a relation which may be defined as the ordinal product of the
relations composing C'P; in this way we get an infinite ordinal product