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

414

RELATION-ARITHMETIC

[PART IV

*17014. F:. a,3 e Cl'C'P.: a - P,1/.. a- 3 C P"(3 - a)
[*1701. *24-55]
*170-141. F:. a, 3e Cl'C'P. D: a - Plo. =. - a C P"(a - /3)
[*170-14-101]
*17015. F: aPCi3. 3 83 n p'P"(a - 3) C a
Dem.
F. *40'12.: F -: y e a -. ). 'P"(a -/3) C Py.
[*22-48]. p np pP"(a - /3) C / n P'y:
[,22-44] D F:y a- 1.,3 PyCa../3n r p'P"(a- /3) C a:
[*10-11-23] D: (y).yea-/./3 n P'y C a.. D, pYp (a-O)Ca (1). (1). *170-11. ) F. Prop
*170-16. F: aC CP. Ca. 3 a. ). aP1/3
Dem.
F.,24-6. 3 F: Hp. D. 2! a- / (1)
F. *243. D: Hp..,8- a = A.
[*37-29]. P(/3- a)=A.
[*24-101-26] D. a- - P3-"( - a) = a- (2). (1). (2). *1701. D F. Prop
*170'161. F: a C'P. C a.. ta. D./P,,a
Dem.
F. *170'16. D F: Hp. D. a (P)C /3.
[*170-101] ). /PIa: D F. Prop
*170-17. F.PI CJ. P J
Dem.
F. *1701. D F: aPc3. D.! a -/.
[*24-55.*22-42] D.a /3.
[*5011] D. aJ13: F. Prop
In order that PdC should be serial, we need further that it should be transitive
and connected. Pd1 is transitive if P is transitive and connected. But Pc1 may
still not be connected: there may be many distinct families in its field, though
all of them must begin with C'P and end with A. For example, if P is a
regression, the class which takes every odd member does not have either of
the relations PC,1 PC1 to the class which takes every even member. In order
that Pc1 should be serial, we require that P should be not only serial, but
well-ordered, i.e. that every existent sub-class of C'P should have a first term.
When P is serial but not well-ordered, P,1 will, however, generate various
series contained in it by imposing suitable limitations on the field.