3
$\begingroup$
  1. $|A|^{|B|} = |A^B|$ ? (cardinal exponentiation)
  2. 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?
  • 0
    1.$|\alpha|^{|\beta|}$ (Ordinal exponentiation).2012-06-06
  • 0
    2.$|\alpha|^{|\beta|}$ (Cardinal exponentiation)2012-06-06
  • 0
    Are they equal?2012-06-06
  • 5
    No, they aren't ... $|\omega^\omega| = \omega$ (ordinal exponentiation) ...2012-06-06
  • 1
    There 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

1 Answers 1