# Initial Segment of Ordinal is Ordinal

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## Theorem

Let $S$ be an ordinal.

Let $a \in S$.

Then the initial segment $S_a = a$ of $S$ determined by $a$ is also an ordinal.

In other words, every element of a (non-empty) ordinal is also an ordinal.

## Proof

By Subset of Well-Ordered Set is Well-Ordered, $S_a$ is well-ordered.

Suppose that $b \in S_a$.

From Ordering on Ordinal is Subset Relation, and the definition of an initial segment, it follows that $b \subset a$.

Then:

\(\ds \paren {S_a}_b\) | \(=\) | \(\ds \set {x \in S_a: x \subset b}\) | Definition of Initial Segment | |||||||||||

\(\ds \) | \(=\) | \(\ds \set {x \in S: x \subset a \land x \subset b}\) | Definition of Initial Segment | |||||||||||

\(\ds \) | \(=\) | \(\ds \set {x \in S: x \subset b}\) | as $b \subset a$ | |||||||||||

\(\ds \) | \(=\) | \(\ds S_b\) | Definition of Initial Segment | |||||||||||

\(\ds \) | \(=\) | \(\ds b\) | as $S$ is an ordinal |

The result follows from the definition of an ordinal.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.6$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals: Theorem $1.7.6$