> Nowhere in the constructions of the real numbers do we make any > assumptions that we are in some exotic set theory with supernatural > powers. This is not X-Men mathematics! The set T which I contructed > in my original post satisfies all the axioms of ZF...
Up to this point you are correct.
> ...and therefore the contructions of the reals work fine therein.
This is where your fallacy lay. Just because all the sets so contructed are in ZF does not imply the other way around. You seem to be adding the following axiom:
NA (The Nathan's Axiom): All sets are finitely constructible.
Theorems of ZF + NA are not the same ones from ZF alone. From ZF + NA, you can deduce that there are only countably many reals (since most transcendental reals will not exist in ZF + NA). However, this statement is false in ZFC.
There is a branch of study called Constructible Set Theory (V=L), which is similar to what you have, except it restricts sets to be constructible (but not necessarily finitely). In this universe, both the Axiom of Choice (AC) and the Continuum Hypothesis (CH) become theorems. In your ZF + NA, the AC remains a theorem, but CH fails since there are only countably many sets in your theory. You may be interested in checking this out: