I think that the following problem originated in a probability textbook :
You have a countably infinite supply of numbered balls at your disposal. They are all labeled with the natural numbers {1,2,3,…}. At 11h30, you put in the urn the balls labeled 1 to 10, and then right after remove the ball labeled 1. Then, at 11h45, you put into the jug balls 11 to 20, and remove the ball number 2, etc. In general at $\frac{1}{(2^n)}$ hours before midnight, you put in balls $(n-1)10+1$ to $10n$, and remove the ball labeled $n$.
The question is : how many balls are left in the urn at (or after) midnight? (that is, after a countably infinite number of steps).
I know that the most accepted answer to this question is that at midnight there aren't any balls left in the urn, because if you consider ball $n$, you know that it has been removed from the urn $\frac{1}{(2^n)}$ hours before midnight, and hasn't been put back at any subsequent step. Thus there can't be any balls in the urn at midnight.
I know that the fact that this problem is very counter-intuitive is probably that it is not worded in any particular axiomatic system, so you can have multiple interpretations of the answer. I know that the above reasoning implies that after midnight, there can't be any balls in the urn that have a natural number of them, and I also know that there are only balls with natural numbered labels available, so that would imply that the jug is empty, but it still is not that much convincing.
consider the following different problem : At the start, there is one ball in the jug labeled 1. Then, at the first step, you remove ball 1 and put in ball 2 at the same time. Then, you remove ball 2 and put in ball 3, etc… There is a ball in the jug at any time, so certainly there should still be ball in the urn at midnight, but it can't have a natural-numbered label.
Consider also the following : in the previous problem, after removing ball #2 at the second step, but back in ball #1. Then at the next step remove ball 1 and put in ball 2, etc. At midnight there is no way to know if the ball is labeled 1 or 2. (probably because the limit of the series $(-1)^n$ doesn't exist?)
My question is : Is there any more satisfying way to formalize this problem, or to explain it that would resolve the paradox that the number of balls in the jug gets constantly larger, never diminishes, and that at the end there is nothing left in the urn?
Best Answer
A good way to formalize the problem is to describe the state of the jug not as a set (a subset of N), because those are hard to take limits of, but as its characteristic function. The first "paradox" is that pointwise limits are not the same as uniform limits. Later ones can be "resolved" by extending the notion of "limit" in various ways, most of which were invented by Euler. See for example the book Divergent Series by Hardy.