I agree that by the time you start assuming ZFC, you're well past the realm of philosophy. The flip side of it is that in a discussion of philosophy you shouldn't make assertions based on axioms that have not yet been agreed on.

In any constructivist axiom system, the usual constructions of the real numbers do not define anything sensible. If you try to make sense of Cauchy sequences from a constructivist point of view, then you'll necessarily wind up with a definition that is very much like the one that I gave.

So you get the result that I stated. From a constructivist point of view, the usual "constructions" of the real numbers are nonsense and construct nothing, while the definition that I gave constructs something that can be reasonably called the real numbers. According to classical mathematics, both sets of constructions are well-defined, and the one that I gave is both complicated and different than the usual reals.

Which version you accept depends on your philosophical beliefs. Seeing what a different philosophical belief will lead to is very hard the first time you do it. But if you believe that it makes no sense to talk about the "truth" of unverifiable statements, you won't wind up debating the finer points of whether to accept C in ZFC...