[SOLVED] Cantor normal form, uniqueness

Nov 2008
23
0
Every ordinal \(\displaystyle \alpha > 0\) can be written uniquely in the form of a finite sum of non-increasing powers of \(\displaystyle \omega\), \(\displaystyle \alpha = \omega^{\beta_1} + \omega^{\beta_2} + \cdots \omega^{\beta_s}\) where \(\displaystyle \beta_1 \geq \beta_2 \geq \cdots \geq \beta_s\), or equivalently, \(\displaystyle \alpha = \omega^{\beta_1} \cdot n_1 + \omega^{\beta_1} \cdots n_2 + \cdot + \omega^{\beta_t} \cdot n_t\) where \(\displaystyle \beta_1 > \beta_2 > \cdots > \beta_t, n_i< \omega, n_i \not = 0\).

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

I do not see how to use the hint to prove this. I would appreciate some hints on this. Thanks.
 
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