TITLE: $\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?
QUESTION [10 upvotes]: Why does $\bigcup \emptyset = \emptyset$ but $\bigcap \emptyset$ is not defined?  
If I had to guess, I'd say it's also equal to $\emptyset$.

REPLY [4 votes]: In some contexts $\displaystyle\bigcap_{\alpha\in\varnothing} S_\alpha$ is defined to be the universal set --- the set that contains everything as a member.  The whole space.
The reason it's undefined in Zermelo–Fraenkel set theory and similar contexts is that there is no set containing the whole space.
The idea behind this definition is that in expressions like
$$
A\cap B\cap C\cap\cdots\cdots,
$$
every term imposes further limits one what can be a member of the set that is the intersection.  If there are no terms, then there are no such limits, so everything is a member.
This is like the fact that $\inf\varnothing=+\infty$.  Every member of a set $S\subseteq\mathbb R$ imposes a limit on how big the infimum can be: it cannot be bigger than that number.  If there are no members of $S$, then there are no such limits.  The infimum is always the biggest it can be subject to those constraints, so when all such contraints vanish, then the infimum is infinite.  This is useful in one context I can think of: the distance between two points in a manifold is the infimum of the set of all lengths of paths connecting them.  Hence if no paths connect them, then the distance is $+\infty$.
I heartily endorse Ilmari Karonen's point.