First, I do think that modelling is in the realm of physics, biology, etc. Mathematicians might make unambiguous statements about these, but they never claim that the model is a good representation of reality.

Second, Godel's incompleteness theorem is about incompleteness, not ambiguity. The statement x = y + 10 is unambiguous in the sense that all mathematicians would always interpret it in exactly the same way. There is no question that x = 12, y = 2 is consistent with that statement, and x=y is not.