Billiards balls solution

See the original puzzle, "The infinite bag of billiards balls".

I feel kind of bad because nobody is actually solving the puzzles that I post. At least, no one but lurkers. Are they too hard, or not interesting enough? Oh, well, I understand that not everyone's into solving puzzles. Well, the solution to this one should be interesting, even if few people are interested in solving the puzzle themselves.

This puzzle is what Wikipedia calls the "Balls and Vase Problem". The paradox is that there are two ways to analyze the problem, each giving different results. On the one hand, every single ball that is put in the bag is eventually taken out. So the bag should be left empty. On the other hand, after the nth step there are 9n balls left in the bag. So as the number n reaches infinity, the number of balls in the bag should also reach infinity. So which is it, zero or infinity?

Well, there is a flaw in one of these approaches, the one that says there are 9n balls after the nth step. How do we prove that there are 9n balls at each step? We use induction. First, we show that it is true for n=1. Then we show if it is true for n=1, it must also be true for n=2, n=3, n=4, and so on. The problem is that induction only works for finite numbers. No matter how many times you increment n, n will never be infinite. It is a mistake to assume that the formula 9n holds true even when n is infinite.

Here is another example of the same mistake. Try to catch where it goes wrong.

1 is a finite number.
If n is a finite number, then n+1 is a finite number.
Therefore, all natural numbers are finite.
Therefore, infinity is finite.

With that out of the way, we can focus on the other approach.

Without some specific information, the resolution to the puzzle is indeterminate. If you put in an infinite number of balls, and take out an infinite number of balls, we might be left with zero, infinity, or even something like twenty-two. However, a definitive answer can be found if we know exactly which balls we remove. For example, if we put in all natural numbers (1, 2, 3, ...) and take out all natural numbers, we're left with nothing. If we instead take out all positive even numbers (2, 4, 6, ...), then we're left with the odd numbers, which is an infinite set.

So you probably can see where this is going. In the main problem, we put in all the natural numbers, and then remove all natural numbers. Therefore, we're left with nothing. In variation 1, we put in all natural numbers, and then remove all numbers that are divisible by 10. We're still left with all the numbers that are not divisible by 10, which is an infinite set.

Variation 2 is the most interesting of all. If you're philosophically inclined, maybe it will provoke questions about how we label objects, and how this relates to their true identity. But I'm not going to dwell on it. In this problem, we are obviously left with an infinite number of balls, since we have not removed a single one. But can you name a single number that is left over? The resolution is that there is not a single finite number left over, but there are many infinite numbers. More specifically, each billiards ball will have an infinite number of zeros in its label. We will have an infinite number of these in the bag.

So, are you tired of thinking about the infinite yet?