There Exists No Universal Set

Theorem
There exists no set which is an absolutely universal set.

That is:
 * $\neg \left({\exists \mathcal U: \forall T: T \in \mathcal U}\right)$

where $T$ is any arbitrary object at all.

That is, a set that contains everything cannot exist.

Proof
Suppose such a $\mathcal U$ exists.

Using the Axiom of Subsets‎, we can create the set:
 * $R = \left\{{x \in \mathcal U: x \notin x}\right\}$

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

Thus:
 * $R \notin \mathcal U$

and so $\mathcal U$ cannot contain everything.