Ordinal Number

Definition

$\mathrm{ord}(A)=o(A)$

Operations

Addition

$A\cap B=\varnothing \Rightarrow o(A)+o(B)=o(A\cup B)$

Associative Property: $(\alpha +\beta )+\gamma =\alpha +(\beta +\gamma )$

Multiplication

$o(A)o(B)=o(B\times A)$

Associative Property: $\alpha (\beta \gamma )=(\alpha \beta )\gamma$

left Distributive Property: $\alpha (\beta +\gamma )=\alpha \beta +\alpha \gamma$

Facts

$\forall A:\exists !o(A)$ where $A$ is a Well-Ordered Set

$\forall a:\exists A,o(A)=a$ where $A$ is a Well-Ordered Set and $a$ is an ordinal number

$|A|=|B|\iff o(A)=o(B)$

$A=\varnothing \iff o(A)=0$

$o(\mathbb{N})=\omega$