Burali-Forti Paradox

Theorem
The class of all ordinal numbers is not a set itself. This is a paradox in naive set theory, which allows comprehension over all classes. That is, symbolically:

{ $x | x$ is an Ordinal } $\notin U$

Let us abbreviate the class of all ordinals to be $\operatorname{On}$

Proof
All ordinals are not elements of themselves, since epsilon is an ordering relation on the ordinal classes. That is:

$\forall A \in \operatorname{On}: A \not\in A$ (1)

However, the class of all ordinal numbers is an ordinal itself. Since for ordering relations on the ordinals, the membership relation is equivalent to the subset relation in all instances (see the definition of Ordinals), we have that:

$\forall x \in \operatorname{On}: x \subset \operatorname{On}$ (2)

The segment of the class of ordinals is:

{ $x \in \operatorname{On} | x \subset \operatorname{On}$ } (3)

Which, by (2) is equal to the $\operatorname{On}$. Therefore $\operatorname{On}$ is an Ordinal.

If $\operatorname{On} \in U$, then it follows that.

$\operatorname{On} \in \operatorname{On}$

But then, by (1) it would follow that:

$\operatorname{On} \not\in \operatorname{On}$ $\perp$

Therefore, by contradiction $\operatorname{On} \notin U$.