There Exists No Universal Set/Proof 4
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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.
$\blacksquare$