## 502 – Ordinal exponentiation

The exercise I mentioned in class is the following: Let $\alpha^{\cdot\beta}$ denote ordinal exponentiation. For ordinals $\alpha,\beta$, define $F(\alpha,\beta)$ as the set consisting of those functions $f:\beta\to\alpha$ such that there are only finitely many $\xi$ such that $f(\xi)\ne0.$

(We haven’t formally defined “finite” yet, but we can take this to mean that the order type of the set $\{\xi\in\beta\mid f(\xi)\ne0\}$ is a natural number, using our formalized notion of natural number.)

For functions $f,g$ in $F(\alpha,\beta)$ set $f\triangleleft g$ iff $f(\xi) for $\xi$ largest such that $f(\xi)\ne g(\xi).$

Then $(F(\alpha,\beta),\triangleleft)$ is a well-ordered set, and its order type is precisely $\alpha^{\cdot\beta}.$