# Thread: [SOLVED] Cantor normal form, uniqueness

1. ## [SOLVED] Cantor normal form, uniqueness

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.