You have an empty urn, and an infinite number of labeled balls.

Almost all people will intuitively feel that the first experiment
(where only balls labeled with multiples of 10 are removed) results in
an urn with an infinite number of balls.

The real excitement starts with the experiment where balls are removed
in increasing order, but 10 times slower than they are added.  Some
feel that the urn will not get empty, due to the slowness of removing.
Some others feel that the urn does get empty, since each ball is
removed at some time during the experiment.  The remaining people claim
that the experiment is not well defined, that it is not possible to do
something an infinite number of times, or something similar,
effectively dismissing the experiment.

Just to put a bit of doubt in some peoples mind, I will add a third experment:

  Let us suppose that at 1 minute to 12 p.m. balls nummbered 1 through 9
  are placed in the urn, and instead of withdrawing a ball we add a zero
  to the label of ball number 1 so that its label becomes 10. At 1/2 minute
  to 12 p.m., balls numbered 11 through 19 are placed in the urn, and we
  add a zero to the label of ball number 2 so that it becomes ball number 20.
  At 1/4 minute to 12 p.m., balls numbered 21 through 29 are placed in the
  urn and ball number 3 becomes ball number 30, and so on.
  At each instant, instead of withdrawing the ball with the smallest label
  we add a zero to its label so that its number is multiplied by 10.
  How many balls are in the urn at 12 p.m. and what are their labels?

If we look at this experiment, at any point in time the inside of the
urn looks exactly like the inside during the execution of the original
paradoxical experiment. However, since no balls leave the urn, it is
now impossible to conclude that the urn will be empty at 12 p.m.
Still, there is no natural number that is the label of any ball in the
urn.  Instead, each ball in the urn will have as its lable a natural
number followed by an infinite number of zero's.

A possible question is now: does this support that the outcome of the
original experiment where balls are removed in increasing order is that
there are an infinite number of balls in the urn? Possibly also with
'infinite natural numbers' as their labels, or are these experiments so
different that the answer is still a clear 'zero'?

I now come to the main points. 
1. Our normal mathematical models do not cater for the COMPLETION of infinite
   tasks (called super tasks by Thomson in 1954).
2. Since we intuitively feel that for many of these experiments there
   are obvious outcomes, we would like to enhance our model to describe the
   outcomes of these experiments.
3. In the enhancement of the model continuity should play an important role.

We include statement 3, since a model in which the conclusion of all
these experiments is that, at 12 p.m. the urn contains "exactly 7
balls, all red" is not desirable, nor useful.

It can be easily shown that general continuity is unattainable. For
instance the sentence "it is before midnight" is true during the
experiment, but is suddenly false after the experiment.

The people claiming that in the second experiment the urn will contain
an infinite number of balls, base this on the fact that the number of
balls in the urn during the experiment, is 9n at (1/2)^(n-1) minute
before 12. They thus assume that this statement is continuous. This
remains to be seen, however.

We have not come to a clear set of criteria which decide whether a
given statement is continuous with respect to performing supertasks. We
did define a "kinematical principle of continuity", which is roughly
formalised as:

        If at some moment before 12 p.m. a ball comes to rest at a particular
        position, which it does not leave till 12 p.m., then it is still at
        that position at 12 p.m.

If we look at the three experiments mentioned, then we can see that in
each case we can come to a conclusion on the contents of the urn.

1. In the first experiment, with the 10-folds being removed, each ball
   which number is a multiple of 10 comes to rest outside the urn (just
   after being removed) and thus is outside the urn at 12 p.m. All other
   balls come to rest inside the urn (just after being placed there), and
   thus are inside the urn at 12 p.m.  Therefore the urn contains an infinite
   number of balls at 12 p.m.

2. In the second experiment, with the balls being removed in increasing order, 
   each balls comes to rest outside the urn. Thus all balls involved are not
   in the urn. Thus the urn is empty.

3. In the third experiment, all balls come to rest inside the urn and thus the
   urn contains an infinite number of balls. The labels of these balls are
   naturall number followed by an infinite number of zero's (since each of the
   numbers is not changed, and zero's once added remain at the label, we can
   draw this conclusion).

The first and third experiment are rather straightforward, while the
second is paradoxical, but not inconsistent.  Please note that is just
one way of extending our model to include super tasks.  We have only
shown that for these experiments, in our model, we come to consistent
conclusions. It does not mean that there are no other models which lead
to different, but also, within that model, consistent solutions.

A final remark: while thinking about these matters, we have wondered
whether we could create a model in which the second experiment would
lead to an urn containing an infinite number of balls. A possibility is
assuming that if a position is continuously occupied by a ball,
although the occupant ball may be swapped every now and again for
another ball, that at 12 p.m. the position is occupied by a so-called
LIMIT BALL.  For the second experiment we could than place balls 1, 10,
100 ..  2, 20, 200, .. each at its own spot in the urn. Each spot in
the urn, once occupied is than continuously occupied with a ball,
leading to limit balls.

This idea of continuity is stronger than the kinematic principle
suggested above, and we have not followed these ideas up enough to
decide whether this extended principle can be made consistent. If any
of the readers have feelings whether this can or cannot be done, I
would be interested to hear their arguments.

I conclude by stating that the result of the super task depends on how
our standard models are enlarged to include the execution of
supertasks. We have given one extension which leads to consistent
results for the supertasks suggested by Ross. Other models may lead to
different, but also consistent, conclusions.

Reference:

Victor Allis and Teunis Koetsier (1991).
On Some Paradoxes of the Infinite.
Brit. J. Phil. Sci. 42 pp. 187-194.

-- allis@cs.rulimburg.nl (Victor Allis)

I am interested in the origin of the puzzle.  As far as I know in this
form the puzzle occurs for the first time in Littlewood's "Mathematical
Miscellanea", which is an amusing little booklet from the 1950s (it may
be even older). Littlewood does not discuss the puzzle.  DOES ANYONE
KNOW OF EARLIER REFERENCES TO THIS PUZZLE?  The puzzle also occurs in
S. Ross's "A first course in probability", New York and London, 1988,
without critical comment.

-- teun@cs.vu.nl (Teun Koetsier)

Every day, put ten balls into an urn and take one out.  After an
infinite number of days, how many balls are in the urn?

If you think there are an infinite number, then here's a proof that
there are none:

Let every ball have a number on it, and let every day have a number.
On the first day (Day 1), put balls 1-10 in and remove ball 1.  On Day
2, put in balls 11-20 and remove ball 2.  In general, on Day n, put in
balls 10n-9 to 10n and remove ball n.

After an infinite number of days, let S be the set of numbers on the
balls in the urn.  If S is non-empty, then it contains a least element
n (because the set of natural numbers is well-ordered).  But ball n was
placed in the urn on Day n/10 (rounded up) and removed on Day n and
never placed in the urn again, so it wouldn't be in the urn after an
infinite number of days.  Therefore, S is empty, and so there are no
balls in the urn after an infinite number of days.

If that proof convinced you, take a second urn and do the following.
On Day n, instead of putting balls 10n-9 to 10n and removing ball n,
put in nine balls labelled 10n-9 to 10n-1, find ball n, and add a 0 to
the end of the label to make it ball 10n.  Now, at the end of every
day, if there is a ball with a label x in one urn, then there will be a
ball with the same label in the other urn, but since you never remove
any balls from the second urn, it will contain an infinite number of
balls at the end of an infinite number of days.

(Confused?  It just goes to show that the natural numbers are really
cruddy for modelling infinite trials.  You should use the surreal
numbers instead, which come up with the "intuitive" result that on day
w = aleph-0, there are an infinite number of balls labelled from w/10
to w.)

-- Matthew Daly <mwdaly@pobox.com>