The

counter-argument, used by

non-mathematicians (and, I must admit, me, when I first heard this

a few years ago) is ``but each step has the

net effect of

adding an

integer number of balls, so surely there should be an

infinite number of balls after an

infinite number of steps''.

The problem with this argument is that it misuses mathematical induction. The argument is basically:

- After step1, there are 9·1 = 9 balls.
- Assume that there are 9
*k* balls after step *k*. Then, after step *k*+1, there will be 9*k* + 10 - 1 = 9(*k*+1) balls.
- Thus, by induction, for any natural number
*n* >=1^{*}, after *n* steps there will be 9*n* balls in the urn.
- Therefore, after an infinite number of steps, there will be an infinite number of balls in the urn.

When stated this way, it becomes relatively

obvious that the error lies in going from step 3. to step 4---it holds for every

natural number, yes, but `

infinity' is not a natural number.

In general, not to sound like a pompous-arsed academic, but your intuitive notions about the nature of the infinite are probably wrong. At least, they do not lead to a consistent mathematical system, and are thus useless to mathematicians.

*****: thus sidestepping the question of whether zero is a natural number.

Think there are an

infinite number of balls left in the urn? Okay, show me a single ball that is in the urn after a

countably infinite number of steps. What is its number?

Putting in nine balls and setting one aside is, when you are dealing with infinite quantities, *not* the same as putting in ten balls and removing the lowest-numbered one. Again, you cannot use common sense when dealing with infinity, at least not if you want an internally consistent system of mathematics.