There’s a math paradox that goes like this:
You have an infinite supply of balls, each labeled with a natural number (1,2,3,4, etc). At step 1, you put the 10 balls numbered 1 through 10 in a (very large) box and then remove the smallest numbered ball in the box, number 1. At step 2, you put the next 10 balls, 11 through 20, in the box and then remove the smallest numbered ball, number 2. At step 3, you put in the next 10 balls (21 through 30) in the box and remove the smallest, number 3. And so on. In the limit, after you have performed step N for all natural numbers N, what is in the box?
By one argument, the number of balls in the box increases with each step (in fact after any step N there are 9*N balls in the box). So there should be an infinite number of balls in the box at the end. But by another argument, every ball has been removed at some step – ball N was removed at step N for every N. So the box will be empty at the end.
Ok, feel free to discuss this paradox if you wish (it seems to generate strong opinions, like the .999… issue), but what I really want to know is what is the NAME of this paradox? My google-fu has failed so far. Perhaps it has already been discussed on this board?