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Let $S\neq\emptyset$ a set. Then $\bigcap S$ exists because it can be written in the form $\{x\in\bigcup S|x\in B~ \forall B\in S\}$ and we only have to use the axiom of union and Schema of Comprehension.

But what happens when $S=\emptyset$? Given that $\bigcup\emptyset=\emptyset$ then we'd have $\bigcap S=\{x\in\emptyset|x\in B~ \forall B\in\emptyset\}$. How do I prove that this set does not exist?

Iesus Dave Sanz
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  • Related: https://math.stackexchange.com/questions/370188/empty-intersection-and-empty-union (see also the questions it points to, especially https://math.stackexchange.com/questions/6613/unary-intersection-of-the-empty-set). – Jean-Claude Arbaut Jan 12 '21 at 23:40

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