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Ordinal Exponentiation

Let alpha and beta be any ordinal numbers, then ordinal exponentiation is defined so that if beta=0 then alpha^beta=1. If beta is not a limit ordinal, then choose gamma such that gamma+1=beta,

alpha^((successor of beta))=(alpha^beta)*alpha.

If beta is a limit ordinal, then if alpha=0, alpha^beta=0. If alpha!=0 then, alpha^beta is the least ordinal greater than any ordinal in the set {alpha^gamma:gamma<beta} (Rubin 1967, p. 204; Suppes 1972, p. 215).

Note that this definition is not analogous to the definition for cardinals, since |alpha|^(|beta|) may not equal |alpha^beta|, even though |alpha|+|beta|=|alpha+beta| and |alpha|*|beta|=|alpha*beta|. Note also that 2^omega=omega.

A familiar example of ordinal exponentiation is the definition of Cantor's first epsilon number. epsilon_0 is the least ordinal such that omega^(epsilon_0)=epsilon_0. It can be shown that it is the least ordinal greater than any ordinal in {omega,omega^omega,omega^(omega^omega),...}.

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References

Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Ordinal Exponentiation

Cite this as:

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

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