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

136

136 ~~~~CARDINAL ARITHMETIC[PRII

[PART III

We then proceed to consider ClS~ arithm. We prove
*115-22. I-:. K E6 Cls3 arith m. sD,: IE Cls2excl: a,86 ie.1!' 20SaAeI'.)'.a,
and *115-23 gives a similar proposition substituting " Prod " for S.
Afte a fw moe prposiions on Cl3aihwe proceed to the associative
law for Prod (*115-34), i.e.
F K IE ClsO arithm. D. Prod'Prod"xK sm Prod'S'K.
(This proposition, *115-34, also states that, with the same hypothesis,
Prod'Is'K sm e&'S'K.) Hence we have
*115-35. 1-: KE C1Cl3 arithm. ). Nc'Prod' Prod"K = Nc'Prod's'K = flNc'IProd'CK
= i-NIe&' E4" = llNc'8s'K
We have also
*115-42. I-: K 6ClCs3 arithm.). Prod'Prodl",c= D""'Prod'E4"K/=D"'cD"cE4'E4"Kf
*115-44. F Cs arithm.:).Prod's'ic =s""Prod'Prod"KC
We have next to prove that if two classes of classes have double similarity,
so have their multiplicative classes. The proof is simple, since the double
correlator is the same as for the original classes, i.e.
*115-502. F: Trs'XeKcgThig-hX.). Trs'Prod'Xe(Prod'K)~nliiiti(Pr-od'X)
whence
*115-51. F: KSMISMX.:).Prod'KsmsrnProd'IX
The number ends with some propositions which result from *114-64-65
and are analogous to them. One of these is used in the following number,
in proving,aff x, Vw = (/,a xc v)'w, namely,
*115-6. F: (R"fy) 1 R, S r yc 1 ---.fy'CG'R.yC OI'S.R"fy, S'fyEc Clslexcl.:).
Prod"R '4y x Prod'S"'y sm e'Cj2 {(2z). z e ry.z = R'"z x S'zI
The subject of this number will be useful in dealing with exponentiation,
since we shall define pY by means of Prod'ca 1,8,", where 1Lt = N,,c'a and
v = N,,c '/.
*115-01. Prod'K= D"6&E'K Df
*115-02. Cls3 arithm = ~ KSKCCs excl) Df
*115-1. F. Prod'K = D~e'ICK [(*115-01)]
*115-101. I-:.a,'K. ),,.wnYal1:WCS'K::). weProd'K [*84-411]
*11.F:: KE6ClS2excl.:D e Prod'K. a:6eK ).n ae wC S'K
[*84-412]
Owing to this proposition, Prod'Kc can be treated without any reference to
IE& K whenever K C ls exel.