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

*200. RELATIONS CONTAINED IN DIVERSITY.

Summary of *200.
Some of the propositions of this number are repetitions or immediate
consequences of previous propositions, especially those of the propositions
of *50 which deal with diversity. But we are chiefly concerned here with
propositions which will be useful in the theory of series; this leads us to
introduce propositions on P"P a and on matters connected with relationarithmetic and other topics. It will be seen that "P2CJ" (i.e. "P is
asymmetrical") is an important hypothesis, as is also Ppo C J, of the use of
which we have already had examples in *96 and *121.
The following are among the most useful propositions in this number:
*200'12.: P e Rl'J. D. C'P. 1c 1
This is the proposition which makes it impossible to define an ordinal
number 1 which shall take its place among relation-numbers applicable to
series.
*200'35. h-:PCJ.ael.D:.Pca=A
This is a consequence of *200'12.
*200'36.: p C J. D. P J
*200 361. F: P2 C J. D. P'x n (tx v P'x) = A. P'x n (P'x v lx) = A
I.e. if P2 C J, no term precedes itself or any of its predecessors, and no
term succeeds itself or any of its successors.
*200-38. F: Ppo C J. D. Ppo = P* n J
— ) 4 -*200-39. F: Ppo G J. x e C'P. * P*'x n tP'x = t'x
We then have a collection of propositions concerned with relationarithmetic.
*200-211. F:PC J.PsmorQ.. Q C J
I.e. the property of being contained in diversity is invariant for likenesstransformations;
*200-4. F: P 4QeR'J. -.P, Qe RI'J. C'P C'Q = A