properties of ordinal arithmetic

Let On be the class of ordinals, and $\alpha,\beta,\gamma,\delta\in\textbf{On}$ . 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 $\alpha\leq\beta$ , then there is a unique $\gamma$ such that $\alpha+\gamma=\beta$
8.

(existence and uniqueness of division): for any $\alpha,\beta$ with $\beta\neq 0$ , there exists a unique pair of ordinals $\gamma,\delta$ such that $\alpha=\beta\cdot\delta+\gamma$ and $\gamma<\beta$ .

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:

•

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

$\omega\cdot 2\neq 2\cdot\omega$ , for the former is $\omega+\omega$ , which consists an element $\alpha$ such that $\beta<\alpha$ for all $\beta<\omega$ , and the latter is $2\cdot\sup\{n\mid n<\omega\}=\sup\{2\cdot n\mid n<\omega\}=\sup\{n\mid n<\omega\}$ , which is just $\omega$ , and which does not consist such an element $\alpha$
•

$(1+1)\cdot\omega\neq 1\cdot\omega+1\cdot\omega$ , 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	2013-03-22 17:51:05
Last modified on	2013-03-22 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