Let
and
be any ordinal numbers, then ordinal exponentiation
is defined so that if then . If is not a limit ordinal,
then choose
such that ,

If
is a limit ordinal, then if , . If then, is the least ordinal greater than any ordinal in
the set
(Rubin 1967, p. 204; Suppes 1972, p. 215).

Note that this definition is not analogous to the definition for cardinals, since
may not equal ,
even though
and .
Note also that .

A familiar example of ordinal exponentiation is the definition of Cantor's first epsilon number.
is the least ordinal such that . It can be shown that it is the
least ordinal greater than any ordinal in .