Definition:Canonical Order

Definition
Let $\On$ be the class of all ordinals.

Let $<$ be the ordinal ordering.

Let $\otimes^l$ be the lexicographic ordering on $\On \times \On$.

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


 * $\tuple {\alpha, \beta} \mathrel {R_0} \tuple {\gamma, \delta}$
 * $\map \max {\alpha, \beta} < \map \max {\gamma, \delta}$ or $\paren {\map \max {\alpha, \beta} = \map \max {\gamma, \delta} \land \tuple {\alpha, \beta} \otimes^l \tuple {\gamma, \delta} }$
 * $\map \max {\alpha, \beta} < \map \max {\gamma, \delta}$ or $\paren {\map \max {\alpha, \beta} = \map \max {\gamma, \delta} \land \tuple {\alpha, \beta} \otimes^l \tuple {\gamma, \delta} }$

Also see

 * Canonical Order Well-Orders Ordered Pairs of Ordinals


 * Initial Segment of Canonical Order is Set


 * Order Isomorphism between Ordinals and Proper Class

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


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