lo.logic - A problem of an infinite number of balls and an urn

You are describing what is known as a
supertask, or
task involving infinitely many steps, and there are
numerous interesting examples. In a previous MO answer, for
example, I described an entertaining example about the
deal with the
Devil, which is similar to your example. Let me
mention a few additional examples here.

In the article "A beautiful supertask" (Mind,
105(417):81-84, 1996), the author Laraudogoitia considers
the situation with Newtonian physics in which there are
infinitely many billiard balls, getting progressively
smaller, with the $n^{th}$ ball positioned at $frac1n$, converging to $0$.
Now, set ball $1$ in motion, which hits ball 2 in such a
way that all energy is transferred to ball 2,
which hits ball 3 and so on. All collisions take place in
finite time, because of the positions of the balls, and so the motion disappers into the
origin; in finite time after the collisions are completed,
all the balls are stationary. Thus:

• Even though each step of the physical system is energy-conserving,
  the system as a whole is not energy-conserving in time.

The general conclusion is that one cannot expect to prove
the principle of conservation of energy throughout time
without completeness assumptions about the nature of
time, space and spacetime.

A similar example has the balls spaced out to infinity, and
this time the collisions are arranged so that the balls
move faster and faster out to infinity (using Newtonian
physics), completing their progressively rapid interactions in finite total time. In
this case, once again, a physical system that is
energy-preserving at each step does not seem to be
energy-preserving throughout time, and the energy seems to
have leaked away out to infinity. The interesting thing
about this example is that one can imagine running it in
reverse, in effect gaining energy from infinity, where the
balls suddenly start moving towards us from infinity,
without any apparent violation of energy-conservation in
any one interaction.

Another example uses relativistic physics. Suppose that you
want to solve an existential number-theoretic question, of
the form $exists nvarphi(n)$. In general, such statements
are verified by a single numerical example, and there is in
principle no way of getting a yes-no answer to such
questions in finite time. The thing to do is to get into a
rocket ship and fly around the earth, while your graduate
student---and her graduate students, and so on in
perpetuity---search for an additional example, with the
agreement that if an example is ever found, then a signal
will be sent up to your rocket. Meanwhile, you should
accelerate unboundedly close to the speed of light, in such
a way that because of relativistic time contraction, the
eternity on earth corresponds to only a finite time on the
rocket. In this way, one will know the answer is finite
time. With rockets flying around rockets, one can in
principle learn the answer to any arithmetic statement in
finite time. There are, of course, numerous issues with
this story, beginning with the fact that unbounded energy
is required for the required time foreshortening, but
nevertheless Malament-Hogarth spacetimes can be constructed
to avoid these issues, and allow a single observer to have
access to an infinite time history of another individual.

These examples speak to an intriguing possible argument
against the Church-Turing thesis, based on the idea that
there may be unrealized computational power arising from
the fact that we live in a quantum-mechanical relativistic
world.