# ordinal exponentiation

Let $\alpha,\beta$ be ordinals. We define $\alpha^{\beta}$ as follows:

 $\alpha^{\beta}:=\left\{\begin{array}[]{ll}1&\textrm{if }\beta=0,\\ \alpha^{\gamma}\cdot\alpha&\textrm{if \beta is a successor ordinal and }% \beta=S\gamma,\\ \sup\{\alpha^{\gamma}\mid\gamma<\beta\}&\textrm{if \beta is a limit ordinal % and }\beta=\sup\{\gamma\mid\gamma<\beta\}.\end{array}\right.$

Some properties of exponentiation:

1. 1.

$0^{\alpha}=0$ if $\alpha>0$

2. 2.

$1^{\alpha}=1$

3. 3.

$\alpha^{1}=\alpha$

4. 4.

$\alpha^{\beta}\cdot\alpha^{\gamma}=\alpha^{\beta+\gamma}$

5. 5.

$(\alpha^{\beta})^{\gamma}=\alpha^{\beta\cdot\gamma}$

6. 6.

For any ordinals $\alpha,\beta$ with $\alpha>0$ and $\beta>1$, there exists a unique triple $(\gamma,\delta,\epsilon)$ of ordinals such that

 $\alpha=\beta^{\gamma}\cdot\delta+\epsilon$

where $0<\delta<\beta$ and $\epsilon<\beta^{\delta}$.

All of these properties can be proved using transfinite induction.

Title ordinal exponentiation OrdinalExponentiation 2013-03-22 17:51:09 2013-03-22 17:51:09 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 03E10 PropertiesOfOrdinalArithmetic