Let ${On}$ be the class of ordinals, and $\alpha,\beta,\gamma,\delta\in {On}$ . Then the following properties are satisfied:

1. (additive identity): $\alpha+0=0+\alpha=\alpha$ (proof)
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 $\gamma<\beta$ .

Conspicuously absent from the above list of properties are the commutativity laws, as well as right distributivity of multiplication over addition. Below are some simple 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 $\beta<\alpha$ for all $\beta<\omega$ , and the latter is $2\cdot \sup \lbrace n\mid n<\omega\rbrace = \sup \lbrace 2\cdot n\mid n<\omega \rbrace =\sup \lbrace n\mid n<\omega\rbrace$ , 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 argument 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.

For properties of the arithmetic regarding exponentiation of ordinals, please refer to this link.