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

72

CARDINAL ARITHMETIC

[PART III

The above definition of exponentiation gives the same value of,v as results
from Cantor's definition by means of" Belegungen." The class of Cantor's
"Belegungen" is
R {R e 1 -- Cls. D'R C a. ('R = },
i.e. (ar T 3)a' /,
and it is easily proved that this is similar to a exp,8.
The usual formal properties of exponentiation result without much
difficulty from the above definitions.
The above definition of exponentiation is so framed as to make propositions on exponentiation independent of the multiplicative axiom, except
when exponentiation is to be connected with multiplication, i.e. when it is to
be shown that the product of v factors, each of which is,u, is pu. This
proposition cannot be proved generally without the multiplicative axiom.
Similarly, in the theory of multiplication, the proposition that the sum of
v t's is pA xc v requires the multiplicative axiom (as does also the proposition
that a product is zero when and only when one of its factors is zero). Otherwise, the theory of multiplication proceeds without the need for employing
the multiplicative axiom.
To take first the connection of addition and multiplication: this connection, in the form in which we naturally suppose it to hold, is affirmed in the
proposition:,e, v e NC. K e V n Cls excl'. D. seC e tL xc (A)
or t, V e NC. K E v n Cl'pu. D. 'K e CL Xc v.
We will take the first of these as being simpler. It affirms that the sum of
v p's is uL x, v. This can be proved when v is s
finite, whether /u is finite or not; but when v
is infinite, it cannot be proved without the
multiplicative axiom. This may be seen as
follows. We know that l. K.
I:, ve NC. ae C.,3 V..____
a "f3 e v n Cls excl',. s'a | ",/ e pu xo v (B)..
K2, s 2
Thus (A) above will result if we can prove
K, X e v n Cls excl',a.. S.s sm s'X,
since we shall put a, "13 for X and use (B). K3 S.
YI 'X

Since K, X e v, we have K sm X. Assume
S e I — +. D'S = K. AIS = X.
Let K1, K2,... be members of K, and let X,,,...
be the members of X which are correlated with
v v
KII, KC2,... by S, i.e. X\ = S',. X2 = S'c. etc. We
have, since KX1, \ e C lK, 1 sm X1. K2 sm X2. etc.

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