Definition:Ordinal Exponentiation

Definition

Let $x$ and $y$ be ordinals.

Ordinal exponentiation $x^y$ is defined using the Second Principle of 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 $0$.

Also see

• Definition:Power (Algebra)

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.30$