Definition 1. An ordinal number or ordinal is a transitive set that is strictly well-ordered by $\in$.
The class of all ordinals is denoted by $\Ord$.
Example. $0 := \empt$ is an ordinal.
Definition 2. For two ordinals $\alpha$ and $\beta$, we define the following total order on $\Ord$:
\[\alpha < \beta \enspace\lrimp\enspace \alpha \in \beta.\]Proposition 1. Every element of an ordinal is an ordinal.
Proposition 2. For any two ordinals $\alpha$ and $\beta$:
- \[\alpha \subseteq \beta \enspace\lor\enspace \beta \subseteq\alpha.\]
- \[\alpha \subset \beta \enspace\rimp\enspace \alpha \in\beta.\]
Proposition 3. For any ordinal $\alpha$,
\[\alpha = \{\beta : \beta < \alpha\}.\]Proposition 4. If $C$ is a non-empty class of ordinals, then $\bigcap C$ is an ordinal and
\[\bigcap C \,=\, \inf C \,\in\, C.\]Proposition 5. If $X$ is a non-empty set of ordinals, then $\bigcup X$ is an ordinal and
\[\bigcup X = \sup X.\]Proposition 6 (Burali-Forti Paradox). $\Ord$ is a proper class.
Proof. By Proposition 5, $\alpha = \sup \Ord$ would be an ordinal, hence
\[\alpha + 1 \leq \alpha,\]a contradiction.