# Ordinal is Transitive/Proof 4

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## Theorem

Every ordinal is a transitive set.

## Proof

Let $\alpha$ be an ordinal by Definition 4.

$\alpha$ is an **ordinal** if and only if:

- $\alpha$ is an element of every superinductive class.

The proof proceeds by the Principle of Superinduction.

From Empty Class is Transitive we start with the fact that $0$ is transitive.

$\Box$

Let $x$ be transitive.

From Successor Set of Transitive Set is Transitive:

- $x^+$ is transitive.

$\Box$

We have that Class is Transitive iff Union is Subclass.

Hence the union of a chain of transitive sets is transitive.

$\Box$

Hence the result by the Principle of Superinduction.

$\blacksquare$

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.7$