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Date: 2-1-2022
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Date: 14-2-2017
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Date: 29-12-2021
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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.
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