Write down the recursive definitions of ordinal addition, multiplication and exponentiation.

Given that $F: \mathbf{O n} \rightarrow \mathbf{O n}$ is a strictly increasing function-class (i.e. $\alpha<\beta$ implies $F(\alpha)<F(\beta))$, show that $\alpha \leqslant F(\alpha)$ for all $\alpha$.

Show that every ordinal $\alpha$ has a unique representation in the form

$$\alpha=\omega^{\alpha_{1}} \cdot a_{1}+\omega^{\alpha_{2}} \cdot a_{2}+\cdots+\omega^{\alpha_{n}} \cdot a_{n}$$

where $n \in \omega, \alpha \geqslant \alpha_{1}>\alpha_{2}>\cdots>\alpha_{n}$, and $a_{1}, a_{2}, \ldots, a_{n} \in \omega \backslash\{0\}$.

Under what conditions can an ordinal $\alpha$ be represented in the form

$$\omega^{\beta_{1}} \cdot b_{1}+\omega^{\beta_{2}} \cdot b_{2}+\cdots+\omega^{\beta_{m}} \cdot b_{m}$$

where $\beta_{1}<\beta_{2}<\cdots<\beta_{m}$ and $b_{1}, b_{2}, \ldots, b_{m} \in \omega \backslash\{0\} ?$ Justify your answer.

[The laws of ordinal arithmetic (associative, distributive, etc.) may be assumed without proof.]