If you want to prove that the universe is an incomplete system that has some axioms that are true but cannot be proven, you have to represent the universe as a system of such axioms, then the use of the theorem would be valid.
Yes and this requires that the universe be fundamentally logical.
In other words every phenomenon in the universe must be describable by some type of mathematical axiom. Such axioms may be totally beyond current human understanding or knowledge but their existence must be possible.
As I said the fact that the universe is ultimately logical is assumed in my argument. I think this is both a reasonable assumption and mostly self evident.
Our limited understanding of quantum mechanics is evidence only of our ignorance and in no way shows that the universe is illogical or indescribable by mathematical axiom.