An ordinal number / ordinal is a transitive set that is strictly well-ordered by $\in$.
The class of all ordinals is denoted by $\Ord$.
For two ordinals $\alpha$ and $\beta$, we define
\[\alpha < \beta \enspace\lrimp\enspace \alpha\in\beta.\]
$0:=\varnothing$ is an ordinal.
PROP-ORD-BF. Burali-Forti Paradox.
$\Ord$ is a proper class.
Every element of an ordinal is an ordinal.
Let $\alpha$ and $\beta$ be ordinals.
- $\alpha\subseteq\beta$ or $\beta\subseteq\alpha$.
- If $\alpha\subset\beta$, then $\alpha\in\beta$.
For any ordinal $\alpha$,
\[\alpha = \{\beta:\beta < \alpha\}.\]
If $C$ is a non-empty class of ordinals, then:
- $\bigcap C$ is an ordinal.
- $\bigcap C\in C$.
- $\bigcap C=\inf C$.
If $X$ is a non-empty set of ordinals, then:
- $\bigcup X$ is an ordinal.
- $\bigcup X=\sup X$.
$\leq$ is a total order on $\Ord$.