# Definition:Ordinal Exponentiation

## Definition

Let $x$ and $y$ be ordinals.

Ordinal exponentiation $x^y$ is defined using Transfinite Recursion:

$\displaystyle x^y = \begin{cases} 0 & : x = 0, \ y \ne 0 \\ & \\ 1 & : x = 0, \ y = 0 \\ & \\ 1 & : x \ne 0, \ y = 0 \\ & \\ \left({x^z \cdot x}\right) & : x \ne 0, \ y = z^+ \\ & \\ \bigcup_{z \mathop \in y} x^z & : x \ne 0, \ y \in K_{II} \\ \end{cases}$

where: