Perhaps I have been unclear, and I'm sorry that Guv ticked me off. I don't think that exchange could have occurred except through a faceless medium like this.

I feel that both sides, zero and infinity, have made compelling arguments for their interpretation. However, I have seen no mathematical argument against either side. The arguments against each side have been logical ones along the lines of "This shows that the answer must be x, and therefore, other answers can't be right." However, both sides are making that argument, and doing so in a manner sufficient to prevent invalidation on a mathematical basis. I say that because there hasn't been such an invalidation yet.

In the early days of math, or so we have been told, there was no 0. That was added, then negative numbers, fractions, decimals, etc. At some point people found that there were non-repeating decimals (e.g. pi) for which there is a name that I forget. However there was still a gap.

At some point, people began accepting calculations using unreal numbers (i) for which there really isn't any logical real-world representation. Surely there were those who felt that using the square root of a negative number in your calculations was absurd.

During the Age of Reason, mathematicians were looking for a "complete" system of mathematics. A complete system was one with the power to define and identify all valid theorems within the system. Godel proved that wasn't possible. No system can be created that can derive all theorems, and once you get close to such a system you can derive at least one theorem that is both true and not true, a paradox.

In math, as in physics, we can find paradoxical results once the system becomes sufficiently close to completeness. The current thread is a similar case, since solid arguments can be made to "prove" two mutually exclusive results.

I am disturbed by this incompleteness we are seeing in some systems. Usually, when we find flaws in the patterns coming from theoretical science, the reason is that our comprehension is inadequate. Being a biologist, I don't think humans are particularly intelligent, since our actions aren't substantially different from other animals, but in this one area, our intelligence is our asset. We must keep moving forward in knowledge. A series of problems of this sort, arising as they do in different fields, suggests that current models can be improved upon. Numbers are only a model of the world around us. We have already expanded that model to inculde unreal numbers, perhaps we need to, again, expand the model.

We know of equations (quadratics) which have two algebraic solutions that are both valid, but are not at all contradictory. Perhaps we will eventually come up with a further set of math to cover equations that have two contradictory solutions. This thread is an example of such a problem. As the number of iterations approaches infinity, the number of balls in the box approaches infinity, but at infinity itself, the number of balls in the box can be either infinite or zero, as many have demonstrated. Why can't there be a further set of numbers beyond unreal.

Perhaps the inconceivable numbers.