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

SECTION B] THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES

129

*114-562.:. Mult ax. D:
(gUS). S 1 - 1. S sm. D'S =. ('S= X... 3 ". sm sm e"X
F. *114-561. *85-61.
F:. (aS). Se 1 - 1. S Gsm. D'S=. (I'S=X.:
e^, eI"X e Cls2 excl: (uYT). Te 1 -* 1. T sm. D'T= eI"K.U'P=eI"X:
[*111-5] D: Mult ax.. e"s sm s e"X:. 3 F. Prop
*114 57. F:. Mult ax. D:
(S). Se 1 - 1. S C sm. D'S = K. IS = X. f. nNc'K= HNc'X
Dern.
F. *114-562 52.
F:. Multax. D: (aHS). Se 1 -- 1. SC sm. D'S = K. IS = X. D.
HNc'e, C I = InNc'e, "X.
[*114-56] D. nNc'K = nNc'X:. D F. Prop
*114-571. F:. Multax.::,, v eNC. K, X e n C'.. INc'K = nNc'X
[*111-52. *114.57]
*114-6. F: K e Cls2 excl. D. Nc'ea'"K = HINc's' K [*85-44]
This is the most general form of the associative law for arithmetical
multiplication.
Owing to the fact that we have two kinds of multiplication, namely
a x 3 and ~a'K, we have four forms of the associative law of multiplication,
namely:
(1) *114-6, above,
(2) *113-54, i.e. F. ( xv) xc w = - x (v xc ),
(3) *114-31, i.e. F: K n X = A. ). IHNc' x, Nc'X = IINc'( v X),
(4) a form of the associative law which has not yet been proved, which
may be explained as follows.
Suppose we have a number of pairs of classes, e.g. (al, /3), (a2, /32),
(a3, /3),.... Suppose we form the products a, x,8, a2 x 832, a3 x 83,... and
multiply all these products together. We wish to prove that (with a
suitable hypothesis) the result is similar to the product of all the a's and
all the 3's taken together as one class; i.e. if we call X the class of products
al x /, a2 x /82, a x /3,..., and p the class whose members are a,, qa, a3,
1, /2,, /3s,..., we wish to prove
nNc'X = lNNc'hL.
In order to express this proposition in symbols, let S be the correlator
of the a's and i3's, so that 3 = S'a,. (The suffix v will not be used further,
since it implies that the number of a's and of /'s is finite or denumerable.)
Then our class of products of the form a x / is

{(ga). a e a a' x S.a},

R. & W. II.

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