Definition:Canonical Order

Definition
Let $\operatorname{On}$ be the ordinal class.

Let $<$ be the ordinal ordering, and let $\operatorname{Le}$ be the lexicographic ordering on $\operatorname{On} \times \operatorname{On}$.

The canonical order on $\operatorname{On} \times \operatorname{On}$, denoted $R_0$, is defined as follows, for ordinals $\alpha, \beta, \gamma, \delta$:


 * $\left({\alpha, \beta}\right) \mathrel{R_0} \left({\gamma, \delta}\right)$
 * $\max \left({ \alpha, \beta}\right) < \max \left({\gamma, \delta}\right)$ or $\left({\max \left({ \alpha, \beta}\right) = \max \left({\gamma, \delta}\right) \land \left({\alpha, \beta}\right) \mathrel{\operatorname{Le}} \left({\gamma, \delta}\right)}\right)$
 * $\max \left({ \alpha, \beta}\right) < \max \left({\gamma, \delta}\right)$ or $\left({\max \left({ \alpha, \beta}\right) = \max \left({\gamma, \delta}\right) \land \left({\alpha, \beta}\right) \mathrel{\operatorname{Le}} \left({\gamma, \delta}\right)}\right)$

Also see

 * Canonical Order Well-Orders Ordered Pairs of Ordinals: $R_0$ strictly well-orders the ordered pairs of ordinals $( \operatorname{On} \times \operatorname{On} )$


 * Canonical Order Initial Segments: Every initial segment of $R_0$ is a set by Canonical Order Initial Segments (unlike the lexicographic order of $\operatorname{On}^2$)


 * Order Isomorphism between Ordinals and Proper Class/Theorem: there exists a function that creates an order isomorphism between $R_0, \operatorname{On}^2$ and $\in, \operatorname{On}$.

We shall call the function constructed by Order Isomorphism between Ordinals and Proper Class/Theorem $J_0$ and we may write:


 * $J_0: \operatorname{On} \times \operatorname{On} \to \operatorname{On}$ is an order isomorphism.