properties of ordinal arithmetic
Let On be the class of ordinals^{}, and $\alpha ,\beta ,\gamma ,\delta \in \text{\mathbf{O}\mathbf{n}}$. Then the following properties are satisfied:

1.
(additive identity): $\alpha +0=0+\alpha =\alpha $ (proof (http://planetmath.org/ExampleOfTransfiniteInduction))

2.
(associativity of addition^{}): $\alpha +(\beta +\gamma )=(\alpha +\beta )+\gamma $

3.
(multiplicative identity^{}): $\alpha \cdot 1=1\cdot \alpha =\alpha $

4.
(multiplicative zero): $\alpha \cdot 0=0\cdot \alpha =0$

5.
(associativity of multiplication): $\alpha \cdot (\beta \cdot \gamma )=(\alpha \cdot \beta )\cdot \gamma $

6.
(left distributivity): $\alpha \cdot (\beta +\gamma )=\alpha \cdot \beta +\alpha \cdot \gamma $

7.
(existence and uniqueness of subtraction): if $\alpha \le \beta $, then there is a unique $\gamma $ such that $\alpha +\gamma =\beta $

8.
(existence and uniqueness of division): for any $\alpha ,\beta $ with $\beta \ne 0$, there exists a unique pair of ordinals $\gamma ,\delta $ such that $\alpha =\beta \cdot \delta +\gamma $ and $$.
Conspicuously absent from the above list of properties are the commutativity laws, as well as right distributivity of multiplication over addition. Below are some counterexamples:

•
$\omega +1\ne 1+\omega =\omega $, for the former has a top element and the latter does not.

•
$\omega \cdot 2\ne 2\cdot \omega $, for the former is $\omega +\omega $, which consists an element $\alpha $ such that $$ for all $$, and the latter is $$, which is just $\omega $, and which does not consist such an element $\alpha $

•
$(1+1)\cdot \omega \ne 1\cdot \omega +1\cdot \omega $, for the former is $2\cdot \omega $ and the latter is $\omega \cdot 2$, and the rest of the follows from the previous counterexample.
All of the properties above can be proved using transfinite induction^{}. For a proof of the first property, please see this link (http://planetmath.org/ExampleOfTransfiniteInduction).
For properties of the arithmetic^{} regarding exponentiation of ordinals, please refer to this link (http://planetmath.org/OrdinalExponentiation).
Title  properties of ordinal arithmetic 

Canonical name  PropertiesOfOrdinalArithmetic 
Date of creation  20130322 17:51:05 
Last modified on  20130322 17:51:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Result 
Classification  msc 03E10 
Related topic  OrdinalExponentiation 