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

[napoleon_in_rags]

42. Agreed, that is the problem, but for both of us.

(Damn, I almost had this post finished, I went to do something else and closed the window. DU needs auto save functionality.)

So basically, yes. .999... is a limit. I write it as
((10^n)-1)/10^n
where ^ is exponent . As each time n increases by 1, another 9 is added to the end. Is it ever equal to 1? No. Roughly, by induction
For numbers positive a and c with a<c, a third number exists b which is 9/10ths of the way between them, which is always less than c.
base case: 9/10 ((10^1)-1)-10^1 < 1.
inductive step: b = (c-a)*(9/10)+a. b < c for all positive a and c with a<c.

Each step of the induction adds another 9 onto the end, and it is never equal to 1. But your saying it gets there at infinity, but it doesn't. The structure of the proof you'd have to make is the same as arguing that there are infinitely long natural numbers. It holds true when the successor function (s(n) = n+1) is called infinitely many times, breaking the inductive step in the proof I made that all naturals are finite above. I don't believe in infinitely long natural numbers, or decimal numbers. I don't think they are proper numbers. But I do think they are both equally as viable.

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.

Oh yeah, that works too. Cool. Also in binary, your numbers are my numbers, but backwards. So the so the set of primes showing the first 3 in mine and yours are:
...101100 = 2^2 + 2^3 + 2^5...
vs.
.001101... = 2^-2 + 2^-3 + 2^-5...

(Where the ellipses in the front is intended to denote an infinitely long natural. )

What I see here though is both of these would require infinite information to denote in full on a computer. They are both just as implausible in this form. What I'm saying is that there is nothing innate about convergence which makes your thing make more sense than my thing, in abstract. Its true that your thing has a decided advantage in terms of magnitudes. For any infinitely long decimal, you can say within a finite amount of steps whether its greater to or lesser than any other (non-equal) one. However, my system makes more sense under other operations. For instance, with the infinite natural

...8761 I can tell you that its remainder divided by 10 is 1, its remainder divided by 1000 is 761, I can tell you its odd, (remainder by 2 is 1) etc. These answers are finite and definitive. Whereas the remainder of pi divided by 10, will be another infinite sequence. If I am willing to get other infinite sequences, I can add, multiply any finite number to any infinitely long number. and in so doing produce a generator that will spill out the results till infinity.

But that's the thing here, what we're really talking about is these generator objects. That's what were doing operations on. For instance with my "set numbers" in binary:
...010101 (even)
+...101010 (odd)
=...111111 (natural numbers)
And the same holds true for your set numbers. We add the generator for all evens with all odds, and get the generator for the naturals . We don't need to expand them to infinity to do operations on them.

This is true because we can see the pattern in the numbers. .01(repeat) is a finite compression of the infinite series .010101... For every well defined number with infinite decimal places, there is such a finite compression, a generator object, specified with finite information which can enumerate its digits forever. A number which cannot be so compressed, consists of an infinite amount of random, uncompressible digits, which can never be specified in any form by human or machine, because to do so would take infinite time. These numbers are uncountably infinite in the consensus thinking, while the numbers specified with finite information are countably infinite. This is really a mess in my opinion: The vast majority of numbers can never be specified, and a particle in space doesn't even at any point in time have a distance from another particle that corresponds to any of these numbers.(Quantum mechanics: At smallest level, particle exists as probability wave until measured and probability wave collapses into 1 of countable measurements) So the vast majority of numbers exist beyond the reach of Man, God, and the Universe. Uh oh.

So in my mind, the coup de grace of the girl in the video was based on the argument that "between any two points on the real line there are an infinite amount of other points" so, because no point can be closer to 1 than .999..., it must be 1. To the extent that idea is right, .999... equal 1. but that idea is not right. I would say:
between any two points on the number line both specified with finite information, there exists a point in between them which takes more information to specify. e.g, in binary:
f(0.1, 1.0) = 0.11
f(0.11, 1.0) = 0.111
f(0.111, 1.10) = 0.1111 and so on.
And, because information is based on the reciprocal of probability, the probability of that number being specified approaches zero as the amount of information needed to specify it grows.
And,
for any arbitrarily large limit on the amount of information that can be expressed by a system, there is some number which IS the closest point to some other number. e.g
For ((10^n)-1)/10^n from above, n corresponds to the amount of decimal digits the system can support. If n is finite, then, then there is always a number which is the closest to 1 and also less than it.

The key thing to remember is that those generator objects aren't numbers, they are like functions. The argument they take is the amount of information available to the system, and they provide the best finite approximation for that system. So the generator object for .999 or the set number of all primes or pi is not equal to any single number, though we can treat them like numbers: Like with Eulers identity, we can relate the generator for e to the generators for sin and cos through i, and show that the generators are the same, so the identity works. But the generator for e still takes as an argument for the precision you are going to calculate it to, and that makes it fundamentally different from ordinary numbers. Of course you can cast ordinary numbers to generators, for instance 1 might be:
f1(m) = (10^m)/(10^m)
where as .999...
f9(m) = ((10^n)-1)/(10^n)
Algrebaically, they are different.