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

348

RELATION-ARITHMETIC

[PART IV

This definition defines the generating relation of the series obtained by
adding x at the end of the P-series; similarly for adding x at the beginning
we put
x<-P=t'xtC'P P Df.
If x is not a member of C'P, the relation-number of P -+ x is the sum of the
relation-number of P and the ordinal 1, which we represent by 1. (The
ordinal 1 has no meaning by itself, but only as a summand.)
The sum of a series of series is defined in the same way as the sum of
two series was defined. Let P be a serial relation whose field consists of
serial relations. Then the sum of all the series generated by members of
C'P, when these series are taken in the order generated by P, must be a
relation which holds between x and y whenever either (1) x and y both
belong to the field of one of the series, and x precedes y in this series, or
(2) x belongs to the field of an earlier series than that to which y belongs.
In the first case, we have (Q). Q e C'P. xQy, i.e. x (Q'C'P) y. In the second
case, we have (IQ, R). QPR. x e C'Q. y e C'R, i.e. (HQ, R). QPR. xFQ. yFR,
i.e. x(F;P)y. Hence the generating relation of the sum of all the series is
s'C'P w F;P. Hence we put
E'P = "'C'P w F;P Df.
The relation U'P has all the properties which we should expect of the sum of
a series of series.
If a series is to result from the addition of a series of series, it is necessary
that no two of the series should have any common terms. For if we have
QPR. xe C'Q n C'R,
we shall also have x (U'P) x.
Hence instead of a series, we shall have cycles; for it is essential to a series
that no term should precede itself. (What seem to be series in which there
is repetition are always the result of a one-many correlation with series in
which there is no repetition, so that a term can be counted once as the
correlate of one term, and again as the correlate of a later term.) For this
reason, as well as for many others, it is important to consider relations
between mutually exclusive relations, i.e. between relations whose fields have
no common terms. We put
A
Rel2 excl = P {Q, R E C'P. Q Q R. GQ,. C 'Q r C'R = A} Df.
Then Rel2 excl has much the same utility in relation-arithmetic as Cls3 excl
has in cardinal arithmetic. We have: R e Rel2 excl. -. FrC'P e Cls -- 1,
which is analogous to the proposition (*84'14)
F-: C e Cls excl -. E e eCls - 1.