Definition:Canonical Order

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

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

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


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

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

However, unlike the lexicographic order of $\operatorname{On}^2$, every initial segment of $R_0$ is a set by Canonical Order Initial Segments.

Therefore, by Order Isomorphism between Ordinals and Proper Class/Theorem, there is 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:


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