# [SOLVED] Cantor normal form, uniqueness

#### zelda2139

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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