Ordinal Exponentiation

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 .

REFERENCES:

Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.

Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.