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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.

Referenced on Wolfram|Alpha

Ordinal Exponentiation

Cite this as:

Weisstein, Eric W. "Ordinal Exponentiation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrdinalExponentiation.html