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

90

CARDINAL ARITHMETIC

[PART III

c and X, T is a double correlator of X and K; that (*111-132) if 5, T are
double correlators of K with X and of X with p respectively, S | T is a double
correlator of K with,u. Hence it follows (*111'45'451'452) that double
similarity is reflexive, symmetrical, and transitive.
We then proceed (*111'2 —34) to consider Crp(S)")X, where it is to be
supposed that S is a correlator of S"X and X, and that S'/ is similar to 8/
if 8 e. We prove
*111-32. F: X, SX e Cls2 excl. S e I 1. Re e'Crp (S)"X. M = s'D'R. ).
M e 1 - 1. ('M= s'X. S"X = MI/"X. S ' X = Me X
Thus in the case supposed, M is a double correlator of S"X and X. Thus
*111'322. F: K, X e Cls2 excl. S e K/ sm X. R e eaCrp (S)"X. M = s'D'R. D.
Me K sm sm X. S=
We then proceed (*1114 —47) to various propositions on "sm sm," and
finally (*111'5'51'53) state three propositions which assume the multiplicative
axiom, namely
*111'5. If K, X e Cls2 excl, then K sm sm X.. [! Ki smi X A Rl'sm.
*111'51. In the same case, 3! K smi X n Rl'sm. D. sK sm s'X, i.e. if K and X
are similar classes of mutually exclusive similar classes, their sums are
similar.
*111*53. In the same case, if K, X e Cls2 excl, c sm sm X. Hence the
multiplicative axiom implies that two mutually exclusive classes of,/
classes each of which has v terms, have the same number of terms in
their sum.
*111-01. K smsmX=(( -1) (I's'X n T(K= Te"X) Df
*11102. Crp (S)'3 = (S'1) -sm 1 Df
*111-03. sm sm = K X (a! K sm sm \) Df
111'1. F: Te,smsmX.. 1 1. aTl'T= s'X. = T [(*111'01)]
1111.: T re sm sm X.. Te (s'cK) sm (s'X). Te X e K sm X
Dem.
F. *3725. Fact. ) F: 'T = s'X. K = Te"X. D. D'T= T"s'. K = Te"X.
[*40-38]. D'T= s'T"'X. K = T"X.
[(*37-04)] ). D'T= s' K (1)
F. *72-451. *60-57. *35-65.: Te l -*. (I'T=s'\.. T Xe 1 1 X = '(T r X) (2). *37-401. ) F: KI= Te"X. =. K = D'(Te X) (3)
F.(1). (2). (3). *4'71.:) T e 1 -- 1. ('T= s't. K = Te"X..
Te -- 1. D'T=sKc. ('T=s'X. Te, Xel —l.D'(Te )=K.tI'(Te rX)=X (4)
F.(4). *1111..73-03. D. Prop