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)