Definition:Ordinal Exponentiation
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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
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.30$