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

412

RELATION-ARITHMETIC

[PART IV

The propositions of this number which will be used in dealing with series
are the following:
*170'1. F: aPcl. -. a,, e CI'C'P. g! a - / - P"(/O - a)
*170'101. F. Pi = Civ'(P)cl
*170-102. F: alP,c/. -. a, e CI'('P.! / - a - P"(a - /)
(These propositions merely embody the definitions.)
*170'11. F:. aP,3. a,, e Cl'C'P:(y). y a-/3./y Ca
This form is often more convenient than *170'1.
*17016. F: aC 'P./3 C a./3 a=.. aPcI
I.e. every sub-class of C'P has the relation Pc1 to every proper part of itself.
*170'17. F. Pc C J. Pi J
*170-2. -:. a, e Cl'C'P: (:ty). y e a -/. P'y n a = P'Y n ': D. aP,,,8
This proposition deals with the case where there is a definite first term y
which belongs to a and not to 3, and whose predecessors all belong to both
or neither.
*170-23. F:. a CP. y a - - P"( - a). D:
y min, (a -,).. P'y n a = P'y n /3
This proposition is useful in case P is well-ordered, since then a - /3 must
have a minimum if it exists (a and / being supposed sub-classes of C'P).
*170-31.: / C C'P. /3 CP. - '. (C'P) P,,3
This follows from *170'16, as does the following proposition:
*170-32. F: a C C'P.! a.. aP1A
*170 35.. Aei = A
*170-38. t::!. ). B'P = C'P. B'Cnv'Pcl = A
*170-6. F: A Pl,8.-. C C'P. a! i
Besides the above, the following propositions should be noted:
*170-36. F. D'Pcl = Cl ex'C'P. E'Pl = CI'C'P - t'C'P
*170-37. F: '! P.. C'Pc = C1'C'P
*170-44. F P smor Q. D. PC1 smor Q,
*170 64. F x ~ e C'P. D. (x <- P)c = (t' u);Pl * Pcl
This proposition shows that every term added to P doubles the number
of terms in PCl; hence it is not surprising that P l (when P is well-ordered)
has a power of 2,. for its relation-number (cf. *177).
*170-67. F: t! P. g! Q. C'P n C'Q = A. ). ('P Q)cI = s;C;(Pc x Qcl)
whence
*170-69. 1: r! P._q! Q. C'P rn C'Q = A. 3.(P 4 Q),1 smor (PQ x Q,)