\begin{definition}[Definition:Lexicographic Order/Ordinals]
The '''lexicographic order''' is a relation on ordered pairs of ordinals denoted $\preccurlyeq_l$.
$\preccurlyeq_l$ is the set of all ordered pairs $\tuple {\tuple {\alpha, \beta}, \tuple {\gamma, \delta} }$ such that:
:$(1): \quad$ Each $\alpha, \beta, \gamma, \delta$ is a member of the ordinal class
:$(2): \quad$ $\alpha \in \gamma$ or $\alpha = \gamma \land \beta \in \delta$
\end{definition}