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

346

RELATION- ARITHIMETIC

[PART IV

*155-42. F: Q smor P.. Nr (Q)'P= Nr'Q
*155-43. F: e NR. ). smor"/ n to '/ =
*155*44. F:. /A, v e NoR.:) = smor"v. -. v = smor"/x
*155-5.. 0r e NoR
*155-51. F. 2r n Rl'Cls e NoR
*155'52. F. 2r n Rel2 e NoR
The following propositions have no analogue in *103.
*155 6. F. C"Nor'P = Noc'C'P
Dem.
F. *100-11. *103-11. FD:. a e Noc'C'P.:
a t'C'P: (aS). Se 1 - 1. D'S= a. ('S= C'P:
[*150'23] -: a e t'C'P: (aS). S e 1 - 1. C'S;P = a. a'S = CP:
[|151'11]: a e t'C'P: (Q). Q smor P. a = C'Q:
[*64-24] -: (gQ) ~ Q smor P. Q e tP. a = C'Q:
[*152'11.*15511] a e C"Nor'P:. D F. Prop
*155'61. F. C"'NoR = N0C [*1556]
On ascending and descending relation-numbers, propositions analogous to
those of *104, *105, and *106 might be proved by proofs analogous to those
given in those numbers. It is, however, scarcely necessary to add anything to
the propositions already proved, namely *154-24-241-242'25'251 on descending
relation-numbers, *154'26-26126262-31 311-32321-322-33-331 on ascending relation-numbers, and *155'23'34 giving the relations of non-homogeneous to
homogeneous relation-numbers. Ascending relation-numbers all exist, and
those that start from the type of P, wherever they end*, are the correspondents' of the homogeneous relation-numbers of the type of P, and are
only some of the homogeneous relation-numbers of the type in which they
end. Descending relation-numbers consist of A together with the homogeneous relation-numbers of the type in which they end: they are the
correspondents of only some of the type in which they begin, or rather, A is
the common correspondent of all those relation-numbers in the initial type
which are not correspondents of any homogeneous relation-number in the
end-type. These properties are exactly the same as in the case of cardinals,
as might be foreseen by *154-14.
* We say that Nr (P)'Q starts from the type of Q and ends in the type of P.
t We call two typically definite relation-numbers correspondents when they only differ as to
the typical determination, i.e. Nr (X)'P and Nr (Y)'P are correspondents.