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

586

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

*91'17.:. P e Potid'R: (]S. Ds. ' (S\ R): 4 (I r CR): D. fP
*91.171. -:. P e Pot'R: OS. Ds. 0 (S | R): R: D. OP
*91'373. F:. P e Pot'R. Dp. UP: -: fR: S ePot'R.. f ). D b(S R)
These are formulae of induction. The first two state that if the property
( is hereditary with respect to R, then if ) belongs to I [ C'R it belongs to
any member of Potid'R, while if p belongs to R it belongs to any member
of Pot'R. The third gives a form of induction which is sometimes more
powerful than the second. It states that if d is hereditary provided its
argument is a power of R, and if )R, then every power of R satisfies (, and
vice versa.
*91.23. F. Potid'R = t(I r C'R) u Pot'R
*91'24. F. Pot'R = i R"Potid'R
These two propositions are very useful as giving relations of Pot'R and
Potid'R.
*91-27. F: P Potid'R. D. C'P C C'R
*91-271. F: P e Pot'R. D. D'P C D'R. (I'P C (R
We do not have in general P e Pot'R )..D'P =D'R. ('P = a'R. If
R is the sort of relation which generates a series (i.e. is either itself serial, or
such that Rpo is serial), the above would characterize a series without a first
or last term. To illustrate the matter, consider a series of four terms,
x, y, z, w, and let R be the relation of immediately preceding in this series.
Thus R holds between x and y, y and z, z and w. Then R2 holds between
x and z, y and w; thus z, which belongs to D'R, does not belong to D'R2.
R3 holds only between x and w; thus neither y nor z belongs to D'R3. All
powers of R beyond the third are null. On the other hand, if we take
a cyclic relation, such as that of left-hand neighbour at a dinner-table,
we shall always have D'P= D'R. IP = (i'R, whatever power of R P
may be.
*91'282. F: P e Pot'R.. P I R e Pot'R
This proposition shows that Pot'R is a hereditary class with respect
to IR.
*91-34.: P, Q Potid'R. D. P Q= Q I P
This proposition states that the relative product is commutative when
each factor is I r C'R or a power of R.
We come next to propositions concerning Rpo. We have
*91-502.. R C RP
*91'504. F. D'Ro = D'R. (P.Rpo = P'R. C'Rpo = C'R