There Exists No Universal Set/Proof 1
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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:
- $R = \set {x \in \UU: x \notin x}$
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