Let $\mathbb S = \varnothing$.
Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$
Consider any $x \in \mathbb U$.
Then as $\mathbb S = \varnothing$, it follows that: $\forall X \in \mathbb S: x \in X$ from the definition of vacuous truth.
It follows directly that: $\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$
That is: $\bigcap \mathbb S = \mathbb U$.
Proofwiki uses the above "proof" to "prove" that intersection of the empty set is the whole universe.
My question is, is the use of vacuous truth really allowed in axiomatic set theory, like ZFC? I don't see how the use of vacuous truth is justified.
The next problem I can think of is that we cannot really "define" the elements of empty set (to my knowledge, there is no element in empty set) so how can we then prove as the above proof did? This seems to contradict the use of vacuous truth.
And of course, there is issue of using the whole universe as a set, and I don't think this is allowed…. (Maybe proof above is using a different axiomatic set theory, as I am using ZF-minded thoughts…)
Best Answer
There’s nothing wrong with the ‘vacuous truth’ part of the argument. It’s perfectly correct that if $X$ is any set, then $\left\{x\in X:x\in\bigcap\varnothing\right\}=X$. To see this, note that if $x\in X$, then $x\notin\bigcap\varnothing$ if and only if there is an $A\in\varnothing$ such that $x\notin A$, and since there is no $A\in\varnothing$ at all, this is not the case.
The problem with the argument is that nothing in $\mathsf{ZF}$ permits the formation of $\left\{x:x\in\bigcap\varnothing\right\}$: this an example of unrestricted comprehension, which is not permitted in $\mathsf{ZF}$. $\mathsf{ZF}$ permits only restricted comprehension, using a formula to pick elements from an already existing set, not from the universe at large.