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. 1.

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

2. 2.

(associativity of addition): $\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$

3. 3.

(multiplicative identity): $\alpha\cdot 1=1\cdot\alpha=\alpha$

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 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 PropertiesOfOrdinalArithmetic 2013-03-22 17:51:05 2013-03-22 17:51:05 CWoo (3771) CWoo (3771) 8 CWoo (3771) Result msc 03E10 OrdinalExponentiation