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

*176. EXPONENTIATION.

Summary of *176.
The definition of exponentiation is framed on the analogy of the definition
in cardinals, i.e. we put
P exp Q = Prod'P Q;Q Df.
We put also, what is often a more convenient form,
PQ0=;(PexpQ) Df.
The relation PQ has for its field (unless Q= A) the class of Cantor's
"Belegungen," i.e. the class (C'P T C'Q),'C'Q. It arranges these by a form
of the principle of first differences, namely as follows: Suppose M and N
are two members of (C'P T C'Q)a'C'Q, and suppose there is in C'Q a term y
for which the M-representative (M'y) precedes the N-representative (N'y),
i.e. for which (M'y) P (N'y), and suppose further that all terms in C'Q which
are earlier than y, i.e. for which zQy. z + y, have their M-representative and
their N-representative identical; in this case we say that M has to N the
relation PQ. This may be stated as follows, provided we assume that P and Q
are series: Let M and N be two one-valued functions whose possible arguments
are all the members of C'Q, while their values are some or all of the members
of C'P. Then we say that M has to N the relation PQ if the first argument
for which the two functions do not have the same value gives an earlier value
to M than to N. Thus for example let P be the
a1 a2 a1 3 a a0
series al, a2, as, a4, a5, and let Q be the series * *.. -p
bl, b2, b3, b4. Then M and N are to be such that
M'b or N'b is defined when, and only when, b is * * * *b, or b, or b3 or b4, and the value of M'b or N'b is bl b2 b3 b4
al or a2 or a. or a4 or a5. Then if M'b, = a, and N'bj $ a1, M precedes N; if
M'b = N'b = a, and M'b2= a. N'b2+ a, M precedes N; and so on. Thus
in this case the first term of the series generated by PQ is the one for which
MJ'b=a, when b has any of the values bl, b2, b3, b4. Thus the first term of
the series is t'al, C'Q, i.e. t'B'P T C'Q. The next term will be
t'a, t (t'bj u tb u tb3) i t'a2 t t'b4,
i.e. L'B'P D'Q u 2p 1 B'Q.