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:
 * $K_{II}$ is the class of all limit ordinals
 * $0$ denotes the zero ordinal
 * $1$ denotes the ordinal $1$, that is: $0^+$, the successor of $0$.

Also see

 * Definition:Power (Algebra)