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

406

RELATION-ARITHMETIC

[PART IV

a selected class / before another selected class v if, where they first differ, /L
chooses an earlier term than v. But if the series P is not well-ordered-if
it is (say) of the type Cnv"w (cf. *263)-there may be no first member of
the field of P where / and v differ. This will happen, for example, if P
consists of all the first terms, and v of all the second terms. Our ordering
relation can be so defined as to put a/ before v in this case also, but if it is so
defined, the associative law of multiplication only holds if P is well-ordered.
For this reason, we define our ordering relation so that, in such a case, /L
comes neither before nor after v. Again, if P is not a Rel2 excl, a member of
a selected class may occur twice, once as the representative of C'Q, and once
as that of C'R, if C'Q and C'R have terms in common. We wish to distinguish these two occurrences. Hence we proceed as follows: If / and v are
two selected classes of C"C'P, let there be one or more members of C'P in
which the /-representative precedes the v-representative, and which are such
that, among all earlier* members of C'P, the j/-representative is identical
with the v-representative.
But a further modification is desirable in order to meet the case in which
two or more of the members of C'P have the same field. Suppose, for
example, we had to deal with a series consisting of all the series that can be
formed out of a given set of terms: in this case, we should have to distinguish
occurrences of any given term not by the field, but by the generating relation.
This requires that we should make an F-selection from C'P, not an e-selection
from C"C'P. Hence we take two members of FI'C'P, say M and N, and
we arrange them or their domains on the following principle: We put
M before N (or D'M before D'N) if there is a relation Q in the field of P
such that the M-representative of Q, i.e. M'Q, has the relation Q to the
N-representative of Q, and such that, if R is any earlier member of C'P, then
M'R is identical with N'R. That is, M precedes N if
(Q): (M'Q) Q (N'Q): RPQ. R $ Q. )R. M'R = N'R.
The relation between M and N so defined has the properties required of
an arithmetical product; hence we put
I'P = MN M, Ne F'C'P:.
(aQ): (M'Q) Q (N'Q): RPQ. R + Q. R. M'R = N'R} Df.
This relation is the ordinal analogue of eIcK. The ordinal analogue of
Prod'K is the corresponding relation of the domains of M and N, i.e. D;WI'P;
hence we put
Prod'P = D;TI'P Df.
In case P is a Rel2 excl, we have Nr'Prod'P = Nr'I'P. But when P is
not a Rel2 excl, Prod'P and II'P are in general not ordinally similar. We
can, however, always make a Rel2 excl by replacing the members x, y, etc. of
* Here Q is said to be earlier than R if Q has the relation P to R and is not identical with R.