Write down the recursive definitions of ordinal addition, multiplication and exponentiation. Prove carefully that $\omega^{\alpha} \geqslant \alpha$ for all $\alpha$, and hence show that for each non-zero ordinal $\alpha$ there exists a unique $\alpha_{0} \leqslant \alpha$ such that

$$\omega^{\alpha_{0}} \leqslant \alpha<\omega^{\alpha_{0}+1} .$$

Deduce that any non-zero ordinal $\alpha$ has a unique representation of the form

$$\omega^{\alpha_{0}} \cdot a_{0}+\omega^{\alpha_{1}} \cdot \alpha_{1}+\cdots+\omega^{\alpha_{n}} \cdot a_{n}$$

where $\alpha \geqslant \alpha_{0}>\alpha_{1}>\cdots>\alpha_{n}$ and $a_{0}, a_{1}, \ldots, a_{n}$ are non-zero natural numbers.

Two ordinals $\beta, \gamma$ are said to be commensurable if we have neither $\beta+\gamma=\gamma$ nor $\gamma+\beta=\beta$. Show that $\beta$ and $\gamma$ are commensurable if and only if there exists $\alpha$ such that both $\beta$ and $\gamma$ lie in the set

$$\left\{\delta \mid \omega^{\alpha} \leqslant \delta<\omega^{\alpha+1}\right\}$$