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Math Help - Cantor Normal Form

  1. #1
    Newbie
    Joined
    Oct 2009
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    Cantor Normal Form

    Hello,

    I'm working on a proof about Cantor Normal Form. In our notes we have that \alpha \omega = \omega ^{\gamma_1} \omega and as part of the proof I think I need to find a similar equation of the form \alpha^{n-1} \omega = ..

    Can anyone suggest how I would find finish this equation?

    Just to put it in perspective, this is the question I'm doing:

    Let \alpha be a limit ordinal with Cantor Normal Form given by \alpha = \omega^{\gamma_1} n_1 + \omega^{\gamma_2} n_2 + ... + \omega^{\gamma_k} n_k and n < \omega. Prove that \alpha^n = \omega^{\gamma_1(n-1)} \alpha .

    EDIT: sorry I should specify that  \alpha, \omega and \gamma are ordinals and n is a positive integer.
    Last edited by whatisthisfor; November 22nd 2009 at 04:59 AM.
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Paris
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    Hi

    Use the associativity of ordinal multiplication!
    If n>1,\ \alpha^{n-1}\omega=\alpha^{n-2}(\alpha\omega)=...
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