Definition:Canonical Order

Definition
Let $R_0$ be a relation on $\operatorname{On} \times \operatorname{On}$. It is called the canonical order on ordered pairs of ordinals. The defining property of the relation is that:


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

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.