Proof of the Second Incompleteness Theorem
The set of axioms produces statements. Some are decidable, some are undecidable. To prove in full range the consistency of mathematics is to prove the validity of all statements, including undecidable ones. Latter to do is impossible by definition. Thus, it is not possible to prove, that mathematics is consistent.
A set of axioms by itself doesn't produce statements (at least not in a way that is relevant to Gödel's theorems), you need a full
formal system to do that, with axioms, an alphabet, a grammar, and rules of inference. What is decidable within such a system depends entirely on the system itself. A statement being undecidable is not a property of the statement, it depends entirely on the formal system in which it is expressed.
Hence you have gone no way at all towards proving Gödel's theorem. It was not known, before Gödel's proof that it was impossible to capture all of number theory within a formal system in which every statement was either true or false (completeness). The necessary of undecidable statements in a consistent, recursively axiomatizable formal system, that is complex enough to encompass number theory, was a consequence of his proof.
You also seem to have confused consistency with completeness.
By the way you can
always construct a formal system in which
any statement you want is true and if you have an
inconsistent formal system, then
every statement can be proved within it.