Definition:Ordinal Exponentiation

Definition
Let $x$ and $y$ be ordinals. If $x \ne 0$, $x^y$ is defined using Transfinite Recursion:


 * $\displaystyle x^\varnothing = 1$
 * $\displaystyle x^{z^+} = ( x^z \cdot x )$
 * $\displaystyle x^y = \bigcup_{z \in y} x^z$ for limit ordinals $y$

If $x = 0$ and $y \ne 0$, then $x^y = 0$ where $0$ denotes the ordinal zero.

If $x = 0$ and $y = 0$, then $x^y = 1$ where $1$ denotes the successor of $0$.