# Initial Segment of Canonical Order is Set

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## Theorem

Let $R_0$ denote the canonical ordering of $\left({\operatorname{On} \times \operatorname{On} }\right)$.

Then, for all $\left({x, y}\right) \in \left({\operatorname{On} \times \operatorname{On} }\right)$, the $R_0$-initial segment is a set.

## Proof

Let $z = \max \left({x, y}\right)^+$.

Let $\left({v, w}\right) R_0 \left({x, y}\right)$.

Then:

\(\displaystyle \max \left({v, w}\right)\) | \(\le\) | \(\displaystyle \max \left({x, y}\right)\) | |||||||||||

\(\displaystyle \) | \(\lt\) | \(\displaystyle z\) |

Thus, the initial segment:

- $\left({\operatorname{On} \times \operatorname{On} }\right)_{\left({x, y}\right)} \subseteq \left({z \times z}\right)$

By Axiom of Subsets Equivalents, the initial segment of $\left({x, y}\right)$ is a set.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.56 \ (2)$