Initial Segment of Canonical Order is Set

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Theorem

Let $R_0$ denote the canonical ordering of $\paren {\On \times \On}$.

Then, for all $\tuple {x, y} \in \paren {\On \times \On}$, the $R_0$-initial segment is a set.

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Proof

Let $z = \map \max {x, y}^+$.

Let $\tuple {v, w} \mathrel R_0 \tuple {x, y}$.

Then:

\(\ds \map \max {v, w}\)

\(\le\)

\(\ds \map \max {x, y}\)

\(\ds \)

\(\lt\)

\(\ds z\)

Thus, the initial segment:

$\paren {\On \times \On}_{\tuple {x, y} } \subseteq \paren {z \times z}$

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By Axiom of Subsets Equivalents, the initial segment of $\tuple {x, y}$ is a set.

$\blacksquare$

Sources

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