There Exists No Universal Set

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
such a $\UU$ exists.

Using the Axiom of Specification, we can create the set:
 * $R = \set {x \in \UU: x \notin x}$

But from Russell's Paradox, this set cannot exist.

Thus:
 * $R \notin \UU$

and so $\UU$ cannot contain everything.