Existence of Set of Ordinals leads to Contradiction

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 from hypothesis that $\On \in \On$.

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

But this is a contradiction.

Therefore a paradox has been formed.

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.