<most of rant snipped> > is countable. Let r be a real number. By our above observations, r is > just some ZF set, and hence r in S_i for some i, hence S in T.
Your "hence" is a false deduction. You have not shown that r is S_i for some i. You seem to be assuming that T exhausts all ZF sets, which is not necessarily true. All you have is T containing all sets which can be constructed in a finite number of steps. You have not disproven the existence of sets which cannot be constructed in this matter.
You appear to be confusing the theory (ZFC) with your model (R). You realize that ZFC does have both countable and uncountable models, don't you?