An old chestnut goes as follows:

I have an infinitely large bin, initially empty, and an infinite supply of balls. At 1/n hours before 12:00, for n = 1, 2, 3, ..., I put 10 balls in a bin, then take one ball out of the bin. At 12:00, how many balls are in the bin?

A closely related problem is this one:
I do the same as the above, but I take that action in each of 2 bins and the balls are numbered 1 to infinity. At 1 hour before 12:00, I put 10 balls numbered 1 to 10 in each bin, and I remove number 1 from the first bin and remove number 10 from the second bin. I continue in the same fashion: At 1/n hours before 12:00, I put balls numbered 10n-9 to 10n in each bin, but I take out ball n from the first bin and ball 10n from the second bin. At the end, which balls are in each bin? What is the total number of balls in each bin? How is this possible?

You probably said that there were infinitely many balls in the bin in the first problem, since at each moment that you take action, the number of balls increases by 9 and never decreases, and you do it infinitely many times.

In the second problem, you could argue that the first bin has just as many balls as the bin in the first problem -- but numbered with every positive integer which is not a multiple of 10 -- while the second bin is empty at the end, since for any specific ball, I can tell you when it was removed from the bin.

This means that the first bin has infinitely many balls, but the second bin has zero balls. Yet, at any given moment, the same number of balls had been added to each bin and the same number of balls had been removed from each bin.

The only flaw here is in treating infinities like normal finite numbers. Infinity does not behave the same way as a number, and you cannot treat it as such and expect to get a sensible answer.

In each problem, you put 10*Infinity balls in the bin, and took Infinity balls out. While it might make sense to you that 10*Infinity - Infinity = 9*Infinity, the truth is that 10*Infinity = Infinity, and Infinity - Infinity is undefined. Infinity - Infinity can come out to any number at all if you manipulate it right, and this problem shows an example of two different manipulations that lead to drastically different results.

Also, to be technically accurate, this "infinity" is aleph null, also written as aleph-null or Aleph-0. There are other infinities larger than aleph null, but you won't run into them easily in simple mathematical calculations.

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