There Exists No Universal Set/Proof 1

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:

$R = \set {x \in \UU: x \notin x}$

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

Thus:

$R \notin \UU$

and so $\UU$ cannot contain everything.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification