If the purported constructions fail, and they do, can we just set up an axiomatic structure that gives us all of the structure in the current theory of the real numbers? No. The theory of complete ordered fields is only decidable with the operations addition and multiplication. As soon as exponents and trigonometric functions are appended to the theory it is no longer decidable. Check out this Wiki link regarding the matter…Decidability of first-order theories of the real numbers.
Quotes…
“Tarski’s exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether this theory is decidable”
“Still, one can handle the undecidable case with functions such as sine by using algorithms that do not necessarily terminate always. In particular, one can design algorithms that are only required to terminate for input formulas that are robust, that is, formulas whose satisfiability does not change if the formula is slightly perturbed.[3]” (my emphasis)
“Alternatively, it is also possible to use purely heuristic approaches.[4]” (my emphasis)
… heuristic approaches… otherwise known as “hand waving”… as in “nothing to be seen here people…keep moving.”
21st Century Paradox 