There Exists No Universal Set/Proof 4

Theorem

There exists no set which is an absolutely universal set.

That is:

$\map \neg {\exists \, \UU: \forall T: T \in \UU}$

where $T$ is any arbitrary object at all.

That is, a set that contains everything cannot exist.

Proof

Aiming for a contradiction, suppose such a $\UU$ exists.

Using the Axiom of Specification, we can create the set of all ordinals:

$\set {x \in \UU: x \text{ is an ordinal} }$

But from Burali-Forti Paradox, this set cannot exist, which is a contradiction.

$\blacksquare$