|A greedy gnome with a countably infinite collection of marbles wants to
trade it with Merlin the mathematician for his beautiful "pearl of
wisdom". Merlin takes advantage of the gnomes unfamiliarity with
the bizarre properties of infinite sets to trick him instead.
The story is very cleverly constructed. We go step by step along with the
gnome. We learn about convergent infinite series (how long it takes him to
number his marbles if he does each in half the time it took him to do the
previous one.) We similarly believe that he can put them all in numbered
boxes in the same way. However, just a tiny twist on this last feat turns
out paradoxically to lead to an unexpected, and unbelievable, outcome.
What this demonstrates, I suppose, is that the physical universe cannot both include an infinite number of things and allow one of those objects to do an infinite number of things to them in a finite amount of time. If those were both the case, then a paradox could be physically achieved. It does not appear that there are an infinite number of particles in the universe, but even if there were (and perhaps there are, despite our current cosmological models) this would not be a problem since the limitation on the speed of travel (e.g. speed of light) would prevent something from acting on two objects at a sufficiently great distance apart in a short amount of time.
(Published in Math Magazine, vol 50, no 3 (May 1977) pp. 141-143.