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

80

CARDINAL ARITHMETIC

[PART III

*110-25. F v E NC. E! sm.~"b. a! s"v ). +e v = s'n ~e SrnS"V
Dem.
F. *10327. ) F: v E NC. a e. v. a! sm., L t. ai! smnI'V.
). = NcOa. v = Nc'6,. {E! smq"111. a,! sm5 "1v.
[*1 ()3-41.*1028."i] ). = N,,c'a. v = N,,cl'. sun = Nc (q)'.
smC"v = Nc (')'fl. a! NC (y)'a. a! Nc (e'fl.
[*110-23] ). + v = Noca ~ +, Noc'/ = sm.q",a I +C sm('66v (1)
F.(1). *10-11-23-.35. )
F 1utL~veNC: ~aE!,a ~ i v.as (2)
F.*37-29. Transp. F: a! sm,",a.!sm j'Ilv.).:!. v (3)
F.(2). (3). DF.Prop
*110-251. F: u,ve NC. ) k0(),vV(1) =tt~+e E
Dein.
F.11025. *104-265. D
F Hp. a! 1.0L(). -j i y0). D. F~0() 0Cy~) = 1u +, v
F. *C110-202. D F +C l i V(~(l) ~ =tl i VI1))~ I ) A 1) ~e v~li = (2)
F. *104-264.) F Hp (2).). (a! a! v).
[* 10-202] D. ~+, v = A.
[(2)] ) (i) ~0 ( - J $+0 (3)
F. (1). (3). D.Prop
*110-252. F p, v E NC. D).puoo0S ~0 v(0,) = A- z' V [Proof as in *110-251-]
A similar proof applies to IL2, (2) etc., and to any stich derived cardinals
whose existence follows from that of kt and v. The proposition does not hold
generally for /wtj, v(,) and other descending (erived cardinals, becaiise they
may be null when,ct and v exist.
The following proposition (*110-3) is more oftei used than any other in
this number except *110-4.
*110'3. F. Nc~a+ Nc c/ N~c'a+,N~c'/3 Nc'(a /3) [*11()022. (* 11003-04)]
*110-31. F: ysma. 8sml. D. Nc' y+eNc=Nc'a+Nc'/3 [*110-24-3]
The following proposition is frequently used.
*110-32. F: arn 13= A. ). Nc'a +0 Nc'fl= Nc'(a v /3) [*110-3-14]
*110-33. F: Nc~a+, Nc6,8.=-.(H7, ). ysinCt.8sm.'y" =A. = YV
[*1 10.3.16]
The above proposition is used in *110-63. We might have use( the above
to define arithmetical addition, but this method would have been less convenient than the method adopted in this number, both because there would