Ordinal Exponentiation via Cantor Normal Form/Limit Exponents

Theorem
Let $x$ and $y$ be ordinals.

Let $x$ and $y$ be limit ordinals.

Let $\langle a_i \rangle$ be a sequence of ordinals that is strictly monotone decreasing on $1 \le i \le n$.

Let $\langle b_i \rangle$ be a sequence of natural numbers.

Then:


 * $\displaystyle \left({ \sum_{i \mathop = 1}^n x^{a_i} \times b_i }\right)^y = x^{a_1 \mathop \times y}$

Proof

 * $\displaystyle \sum_{i \mathop = 1}^n \left({ x^{a_i} \times b_i }\right) \le x^{a_1} \times \left({ b_1 + 1 }\right)$ by Upper Bound of Ordinal Sum

Furthermore:


 * $\displaystyle x^{a_1} \le \sum_{i \mathop = 1}^n \left({ x^{a_i} \times b_i }\right)$

It follows that:

It follows that:

Also see

 * Cantor Normal Form