The following question come from remarks from Velleman, How to Prove it (pp.82, ex. 14).
Let $F$ denote a family of sets and consider the case where $ F = \emptyset$. Then the statement, $ x \in \cap F$ is always true, regardless of $x$. Thus the intersection of an empty set is the universe of discourse.
Paraphrasing from Velleman, the universe of discourse means different things depending upon the context and sometimes no universe of discourse is used. Thus some mathematicians consider $\cap \emptyset$ to be meaningless.
Since only some mathematicians consider it to be meaningless, this suggests that some mathematicians believe it to have meaning.
In what situations may the intersection of an empty set have meaning?