ordinal exponentiation

Let α,β be ordinalsMathworldPlanetmathPlanetmath. We define αβ as follows:

αβ:={1if β=0,αγαif β is a successor ordinal and β=Sγ,sup{αγγ<β}if β is a limit ordinal and β=sup{γγ<β}.

Some properties of exponentiationMathworldPlanetmathPlanetmath:

  1. 1.

    0α=0 if α>0

  2. 2.


  3. 3.


  4. 4.


  5. 5.


  6. 6.

    For any ordinals α,β with α>0 and β>1, there exists a unique triple (γ,δ,ϵ) of ordinals such that


    where 0<δ<β and ϵ<βδ.

All of these properties can be proved using transfinite inductionMathworldPlanetmath.

Title ordinal exponentiationMathworldPlanetmath
Canonical name OrdinalExponentiation
Date of creation 2013-03-22 17:51:09
Last modified on 2013-03-22 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