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Math Help - [SOLVED] Cantor normal form, uniqueness

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    [SOLVED] Cantor normal form, uniqueness

    Every ordinal \alpha > 0 can be written uniquely in the form of a finite sum of non-increasing powers of \omega, \alpha = \omega^{\beta_1} + \omega^{\beta_2} + \cdots \omega^{\beta_s} where \beta_1 \geq \beta_2 \geq \cdots \geq \beta_s, or equivalently, \alpha = \omega^{\beta_1} \cdot n_1 + \omega^{\beta_1} \cdots n_2 + \cdot + \omega^{\beta_t} \cdot n_t where  \beta_1 > \beta_2 > \cdots > \beta_t, n_i< \omega, n_i \not = 0.

    Hint. For the uniqueness, prove first by induction on s that if \beta_1 \geq \beta_2 \geq \cdots \geq \beta_s and \gamma = \omega^{\beta_1} + \omega^{\beta_2} + \cdots \omega^{\beta_s}, then \gamma < \omega^{\beta} + \gamma.

    I do not see how to use the hint to prove this. I would appreciate some hints on this. Thanks.
    Last edited by zelda2139; April 24th 2010 at 06:23 PM.
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