\begin{definition}[Definition:Ordinal Exponentiation]
Let $x$ and $y$ be ordinals.
'''Ordinal exponentiation''' $x^y$ is defined using Transfinite Recursion:
:$\ds x^y = \begin{cases} 
0 & : x = 0, \ y \ne 0 \\
 & \\
1 & : x = 0, \ y = 0 \\
 & \\
1 & : x \ne 0, \ y = 0 \\
 & \\
\paren {x^z \cdot x} & : x \ne 0, \ y = z^+ \\
 & \\
\ds \bigcup_{z \mathop \in y} x^z & : x \ne 0, \ y \in K_{II} \\
\end{cases}$
where:
* $K_{II}$ is the class of all limit ordinals
* $0$ denotes the zero ordinal
* $1$ denotes the ordinal $1$, that is: $0^+$, the successor of $0$.
\end{definition}