ordinal exponentiation
Some properties of exponentiation^{}:

1.
${0}^{\alpha}=0$ if $\alpha >0$

2.
${1}^{\alpha}=1$

3.
${\alpha}^{1}=\alpha $

4.
${\alpha}^{\beta}\cdot {\alpha}^{\gamma}={\alpha}^{\beta +\gamma}$

5.
${({\alpha}^{\beta})}^{\gamma}={\alpha}^{\beta \cdot \gamma}$

6.
For any ordinals $\alpha ,\beta $ with $\alpha >0$ and $\beta >1$, there exists a unique triple $(\gamma ,\delta ,\u03f5)$ of ordinals such that
$$\alpha ={\beta}^{\gamma}\cdot \delta +\u03f5$$ where $$ and $$.
All of these properties can be proved using transfinite induction^{}.
Title  ordinal exponentiation^{} 

Canonical name  OrdinalExponentiation 
Date of creation  20130322 17:51:09 
Last modified on  20130322 17:51:09 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03E10 
Related topic  PropertiesOfOrdinalArithmetic 