- $|A|^{|B|} = |A^B|$ ? (cardinal exponentiation)
- Let $\alpha$ and $\beta$ be ordinals and $\gamma$ = $|\alpha|^{|\beta|}$ (Ordinal exponentiation)
Then is $\gamma$ an initial ordinal(thus cardinal) and can the ordinal exponentiation in this case be understood as a cardinal exponentiation?
Ordinal and cardinal exponentiation
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elementary-set-theory
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01.$|\alpha|^{|\beta|}$ (Ordinal exponentiation). – 2012-06-06
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02.$|\alpha|^{|\beta|}$ (Cardinal exponentiation) – 2012-06-06
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0Are they equal? – 2012-06-06
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5No, they aren't ... $|\omega^\omega| = \omega$ (ordinal exponentiation) ... – 2012-06-06
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1There is one countably infinite cardinal, and uncountably many countablby infinite ordinals. Some of them defined as exponents of others. Hence exponentiation works differently between the two. – 2012-06-06