# Existence of Set of Ordinals leads to Contradiction

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## Theorem

The existence of the set of all ordinals leads to a contradiction.

## Proof

Suppose that the collection of all ordinals is a set.

Let this set be denoted as $\On$.

From Ordinals are Well-Ordered: Corollary, it is seen that $\Epsilon {\restriction_{\On} }$ is a strict well-ordering on $\On$.

By Element of Ordinal is Ordinal, it is seen that $\On$ is transitive.

And so $\On$ is itself an ordinal.

Since $\On$ is an ordinal, it follows by hypothesis that $\On \in \On$.

Since $\On$ is an ordinal, it follows from Ordinal is not Element of Itself that $\On \notin \On$.

This is a contradiction.

$\blacksquare$

## Also known as

This theorem is sometimes presented as the **Burali-Forti Paradox**, but strictly speaking it is in fact a resolution to it.

## Also see

- The class of all ordinals, $\On$
- Russell's Paradox, another paradox in naive set theory.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.13$