One way to treat $\bigcap\emptyset$ In "Elements of Set Theory" by Enderton 25~26p,

Thus it looks as if $\bigcap\emptyset$ should be the class $\mathsf V$ of all sets. ... This presents a mild notational problem: How do we define $\bigcap\emptyset$? ... One option is to leave $\bigcap\emptyset$ undefined, since there is no very satisfactory way of defining it. This option works perfectly well, but some logicians dislike it. It leaves $\bigcap\emptyset$ as an untidy loose end, which they may later trip over. The other option is to select some arbitrary scapegoat (the set $\emptyset$ is always used for this) and define $\bigcap\emptyset$ to equal that object. ...

In explaning the second option, I don't understand "(the set $\emptyset$ is always used for this)". What does it mean? In this case do we use $\emptyset$ instead of $\bigcap\emptyset$? (If so, $\emptyset$ will have two meaning, empty set and $\bigcap\emptyset$) And what does the word "always" mean? Does it mean whenever we have a notational problem, we will always choose $\emptyset$ in order to denote it? (Then $\emptyset$ will have so many meaning, is it really right?)
 A: Option 1: define $\bigcap \emptyset$ to be the class of all sets. This is consistent with the definition of $\bigcap$ elsewhere but somehow still feels weird.
Option 2: define $\bigcap A$ only if $A$ is nonempty.
Option 3: define $\bigcap \emptyset$ to be some particular set by convention. The most natural choice seems to be $\emptyset$. But whatever you pick it to be, it will not be consistent with the definition of $\bigcap$ elsewhere, unless it is the class of all sets.
A: That author gives a pretty weak/handwavy argument for the "scapegoat" convention, in my opinion. Normally it makes much more sense to think of expressions like $\bigcap\emptyset$ as undefined. The scapegoat approach is only necessary in technical situations where "conditionally (un)defined notation" is unavailable or inconvenient, like when we want simple rules for expanding definitions in order to formally reason about a deductive system (or implement it in software).
Either way, you have to be careful that a notational construct's conditions are met when you use it - or if not, at least take care to ensure that the undefined/scapegoat value doesn't affect the truth of the surrounding statement.
