properties of ordinal arithmetic

Let On be the class of ordinals, and $\alpha ,\beta ,\gamma ,\delta \in \text{𝐎𝐧}$. 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 $\alpha \le \beta$, 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).