There Exists No Universal Set

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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 Specification, 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.

$\blacksquare$


Sources