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)<g(\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}.


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