Understanding of convergence of intersections of sets

Question

If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of $\left(\large\frac{-1}{n}, \frac{1}{n}\right)$ as $n\to \infty$)

However, at no time during the sequence is there a first element with only one point. In fact, for any finite (but unbounded) $n$, the number of points in the interval is uncountably large. So, how do you understand the idea that this infinite intersection contains just one point? How do you make sense of it?

Answer 1

A point is in the intersection if it's in every set of the sequence. Focus on that. No matter how close to $0$ a point is (excluding $0$ itself), there will be a sufficiently large $n$ such that the set $(-1/n,1/n)$ does not contain it, and therefore that point will not be in the (infinite) intersection, because, again, a point is in the intersection iff and it is in every set specified by the intersection.

Answer 2

I think the source of the confusion is this: taking the cardinality of a set seems like such a simple operation that we subconsciously expect it to be continuous. We might think that simple limiting operations like countable nested intersections should give convergent cardinalities. But the simple fact is that they don't; our subconscious guess is just wrong, and this example proves it. Cardinality is not continuous in this sense.

This is related to the standard "paradox" of the supertask where at time $1-2^{-n}$ you add to a box two balls numbered $2n-1$ and $2n$, and remove the one numbered $n$. Despite the fact that the cardinality of the balls in the box is always increasing, in the limit at time 1 the box is empty. Again, the paradox arises only if you feel that cardinality should be continuous. If you realize that it isn't, then you no longer see such examples as paradoxical.