Definition:Canonical Order

Definition

Let $\operatorname{On}$ be the ordinal class.

Let $<$ be the ordinal ordering.

Let $\operatorname{Le}$ be the lexicographic ordering on $\operatorname{On} \times \operatorname{On}$.

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

$\left({\alpha, \beta}\right) \mathrel{R_0} \left({\gamma, \delta}\right)$

if and only if:

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

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:

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

Sources

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