How are Sets different from Boxes? I am reading the book
"Naive Set Thoery" by Sir Halmos. There, in
Section 2: Axiom of Specification, pg-5, first line, he writes
"The analogy between sets and boxes has many weak points"
What are these drawbacks? i.e Are there any properties which sets have and boxes do not, or vice-versa?
 A: One big issue is that an object can only be in one box at a time, but it can be in multiple sets.  For instance, the sets $\{1,2\}$ and $\{1,3\}$ cannot simultaneously be viewed as boxes, since the number $1$ would have to be in two different boxes at once.  You could get around this by allowing multiple "copies" of objects, so you have two different $1$s, one in each box.  This creates more problems, though: what if you put both $1$s in the same box?  A box with two $1$s in it is different from a box with one $1$, but $\{1,1\}$ is the same set as $\{1\}$.
Another issue is extensionality: two sets which have the same elements are equal, but is that true of boxes?  If you have two different boxes which each contain a $1$, are they actually the same box?  Well, maybe you say they are, as an extension of our policy of saying the two $1$s are "the same".  But what about two empty boxes?  Surely you can have two different boxes that are both empty, but extensionality means there can only be one empty set.
