If you're having math problems, I feel bad for you, son...

[Dr. Strange]

41. The problem is you can't treat infinity like a real number.

That's the whole reason for calculus and analysis: to come up with a rigorous way of dealing with infinity.

You could just as well say:

.999... = 1- (1/infinity), that latter term is also not equal to zero.

I wouldn't be comfortable saying that, because we're putting infinity into the fraction on the right, which is treating it like a real number. That potentially leads to faulty deductions like infinity/infinity = 1, since x/x = 1 for every positive number.
I would however be comfortable writing
lim_(n->infinity) 9/10 + 9/10² + 9/10³ + ... + 9/10ⁿ = lim_(n->infinity) 1 - 1/10ⁿ
which replaces treating infinity like a real number with the idea of limits. It also gives a justification for 0.999... equaling 1.

To see this, look at the number e defined as a limit:

http://en.wikipedia.org/wiki/E_(mathematical_constant)
And notice that if 1/infinity = zero, than e is equal to 1, which it isn't.

True (which is another reason why we can't treat infinity like a number), but notice what's going on in the limit: inside the parentheses, the 1 + 1/n term is approaching 1--however, the exponent is going off to infinity. So, the two parts (the base and the exponent) are battling for control: does the inside go to 1 faster than the exponent goes to infinity?

now what would a repeating decimal expansion of 1/infinity be? 0.00... repeating zeros forever, with a 1 on the "end" (which you never get to). subtract that from 1 and you get repeating 9's forever.

Since we can't treat infinity like a real number, there would be no decimal expansion for 1/infinity.

The meat of my argument is that a number can't be fully expressed in a number system where it requires infinite representation.

It can, but it does require some caution. The construction of the real numbers from the rational numbers is a large part of what mathematical analysis is all about.

The reason I chose those set numbers is because written in binary, a there is a 1 for each number in the set, a zero for each absent in the set. You can go on iterating them forever from the smallest to larger parts, so in this way they are like decimal representations that go on forever as well, just reversed so the decimals are more useful for talking about magnitude. In the big scheme of things, one doesn't make much more sense than the other. Infinite sets are just like infinitely long decimals or infinite integers. They really exist as objects with finite definitions, and that's where they belong, in the context where they can be finitely defined.

Your construction above only works for finite sets, since like I said, the subsets {2, 3, 5, ...} and {10, 20, 30, ...} would not map to any natural number; however, you can adjust your construction in the following way: map a subset {a₁, a₂, a₃, ...} of the natural numbers (where we might as well assume a₁ < a₂ < a₃ < ...) to the number 2^-a₁ + 2^-a₂ + 2^-a₃ + .... This series will converge, and it gives a 1-1 correspondence between the real numbers between 0 and 1 and the set of all subsets of the natural numbers. But it relies on only allowing "infinite" numbers after the decimal point.