It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1.
And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof relies on the existence of a supremum, which is non-trivial.
I think this is spot on, at least for me personally.
I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers".
Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from:
2 * inf = inf
inf + inf = inf (subtract inf from both sides)
inf = 0
There is this phrase, often used when describing the decimal expansion of pi - "keeps going infinitely". This phrase is not exactly incorrect, but I wonder if it misleads people into thinking that an "infinite decimal" is "a kind of infinity", which it really isn't in any meaningful way.
Infinity, the number, is routinely confused with creating an onto function mapping digits of pi to a set with a cardinality of the natural numbers. But sadly most people don't have the mathematical maturity to understand the difference when they encounter their first irrational number (normally pi).
Multiply both sides by ‘x’, then subtract one side from the other, then take the limit as x -> inf. This is obviously undefined. To get to zero, you have to make a new rule that one form of infinity is bigger than another form of infinity.
Infinity is very slippery, and there are several divergent fields of math that depend on particular definitions of it.
It's true that it depends on the definitions of real/complex numbers. Many other things turn out to be provable from ZFC. A discussion about this, from the viewpoint of Metamath, is here: http://us.metamath.org/mpeuni/mmcomplex.html
And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof relies on the existence of a supremum, which is non-trivial.