# Definition:Canonical Order

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

- $\tuple {\alpha, \beta} \mathrel {R_0} \tuple {\gamma, \delta}$

- 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

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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.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.55$, $\S 7.57$