Definition:Canonical Order

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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 $\alpha, \beta, \gamma, \delta$:

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

Also see

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.