[soVeryTired]

But the axiom of choice is logically independent of the other zermelo frankel axioms. You can assume zermelo franked plus choice, and get the standard framework (ZFC) that let's you build the reals.

But you can also assume ZF plus the negation of the axiom of choice, and get a system that is consistent if and only if ZF is consistent. It's not clear (to me, at least) that this other system will let you build the real numbers.