Burali-Forti Paradox

Theorem

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

Suppose that the collection of all ordinals is a set.

Let this set be denoted as $\operatorname {On}$.

From the corollary of ordinals are well-ordered, it is seen that $\Epsilon {\restriction_{\operatorname {On}}}$ is a strict well-ordering on $\operatorname {On}$.

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

And so $\operatorname {On}$ is itself an ordinal.

Since $\operatorname {On}$ is an ordinal, it follows from hypothesis that $\operatorname {On} \in \operatorname {On}$.

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

But this is a contradiction.

Therefore a paradox has been formed.

$\blacksquare$