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

SECTION D]

ARITHMETIC OF RELATION-NUMBERS

475

The above process is symbolized as follows. The generating relation of
the series (a, Q, b, Q, c 4 Q) is 4 Q;Q; thus the three relations whose sum
is to be taken are, Q;Q, 4 R;R,, S;S, i.e. using the notation of *182,
I, '
according to which we put x = x x, our three relations are 'Q, 4 'RR, 4 'S.
But the generating relation of the series ( 'Q, 4 'R, 1 'S) is,;P, since.,., M,
P= (Q 4 R 4 S). Thus 4;P is the relation required for defining the sum of
tihe relation-numbers of members of the field of P; i.e. we put
2Nr'P= Nr'E' 4;P Df.
We will call 4;P the separated relation corresponding to P. 4;P is constructed, as above, by replacing every member x of C'Q, where Q e C'R, by
x Q; so that if x belongs both to C'Q and to C'R, it is duplicated by
being transformed once into x, Q, and once again into x, R.
For the treatment of products, we do not require,;P, because I'P has
been so defined as to effect the requisite separation. We might, however,
by the use of,;P, have dispensed with I'P as a fundamental notion, and
contented ourselves with Prod'P; for we have
n'P= -;Prod';P.
Thus we might have taken Prod as the fundamental notion, and defined [I
by means of it.
The addition of unity to a relation-number has to be treated separately
from the addition of two relation-numbers, for the same reasons which
necessitate the treatment of P -1 x and.x 4- P separately from P - Q. There
is no ordinal number 1, but we can define the addition of one to a relationnumber. If Nr'P = / and xe.e C'P, we must have
Nr'(P x)=/u + i,
where we write " " for unity as an addendum. We do not write " 1,.,"
because we shall, at a later stage, give a general definition of /pr, in virtue
of which, if' t is an inductive cardinal, Fr will be the corresponding ordinal.
This definition entails lr=A, and therefore we use a different symbol "i"
for 1 as addendum. The symbol i is only defined in its uses, and has no
significance except in a use which has been specially defined.
W\e define the product /JtL v as the relation-number of P x Q, when, = Nor' andl v = N,,rQ. The product so defined obeys the associative
law, and obeys the distributive law in thie form
(P + W) X< I = (P X< ) + (X X p)
but not, in general, in the form
L X< (v +,) = ( X v) i- (LU X <).