ordinal exponentiation

Let $\alpha ,\beta$ be ordinals. We define ${\alpha }^{\beta }$ as follows:

Some properties of exponentiation:

1. $0^{\alpha }=0$ if $\alpha >0$

2. $1^{\alpha }=1$

3. ${\alpha }^1=\alpha$

4. ${\alpha }^{\beta }\cdot {\alpha }^{\gamma }={\alpha }^{\beta +\gamma }$

5. ${({\alpha }^{\beta })}^{\gamma }={\alpha }^{\beta \cdot \gamma }$

6. For any ordinals $\alpha ,\beta$ with $\alpha >0$ and $\beta >1$, there exists a unique triple $(\gamma ,\delta ,ϵ)$ of ordinals such that

   $$\alpha ={\beta }^{\gamma }\cdot \delta +ϵ$$

   where and .

All of these properties can be proved using transfinite induction.