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

638

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

Put I,'x = Rt'x rn z (zRI,oz) Dft
J1 ' = R',x n i {(zRpoz)} Dft
(these definitions being only to apply within *96). Then JR'x is the open
part of the series R*'x, and I'x is the circular part. The open part wholly
precedes the circular part, provided R e Cls - I1; i.e.
R e Cls -:. D. J,.'x C 1'R,,,"I'x.
If JR'x and IR'x both exist, J,'x has a last term, say y. The successor of
4 -this term, R'y, is the only term in R*'x which has two immediate pre4- v -- v
decessors in R*'x, namely y and t'(I1,,x rn R''y).
The most important applications of the propositions of the present
number are in the theory of finite and infinite, both cardinal and ordinal.
When R is many-one, then if IR'x exists, or, more generally, if JR'x has
4 -a last term, R*'x is a finite class, i.e. what we shall call a " Cls induct" (cf.
*120). That is, we have
4-: R e Cls — )1. E! max/'Jj'x.. RD. x e Cls induct.
4 -If JR'x exists, but has no last term, R*'x is a progression (cf. *122) when
its terms are arranged in the order generated by R. That is, giving to 0o
and o the meanings given by Cantor (cf. *123 and *263), and using "Prog"
for the class of one-one relations which generate progressions, we have: R e CIs -- 1. E! maxl?'Jl'x. 3! JJ'x.).
4- 4- 4 -R*'x E o. (R*'x) 1 R e Prog. (R*'x) 1 Ro e o.
Another very important proposition in the proof of which the present
number is useful is *121'47, which proves that if R is either one-many or
4- --
many-one, and a and z are any two terms whatever, then R*'a n R*'z (which
we call the " interval" from a to z) is always a finite class. The proof that
progressions are well-ordered series depends upon the propositions of this
number, since it uses *122-23, which depends upon *96'52.
The present number begins with a series of propositions (ending with
*96'16) on a 1 Rp and a 1 R*, both in general and when a = R*'x. We then
4 -proceed to a few propositions (*96'2 —25) on (R*'x) 1 R when R e - Cls;
with the exception of *96'24, these propositions are all used in the cardinal
theory of finite and infinite. They are, however, less important than the
-t
subsequent propositions, which are concerned with RB'x when R e Cls-, 1.