The perveyors of Gödelian Uncertainty love to cast it like a wet blanket over all of Mathematics, and by extension onto physics; but here's the rub:
They never seem to grasp that just because there exists SOME truths about arithmetic of Natural numbers that can't be proven or disproven from within the system, it doesn't mean that the theorems we DO PROVE are in any way suspect!
So long as Mathematicians, and Physicists, restrict themselves to using only those Mathematical Theorems which HAVE been proven, they can never fall victim to Gödelian Uncertainty. An unproven theorem is still an unproven theorem, regardless of whether it's unproven because it's difficult (Fermat's Last Theorem), or because it is a Gödel statement. And, as long as we don't use unproven theorems, they have no effect on anything we are doing!
In conclusion, Gödel's Incompleteness theorem poses no practical impediment at all to most of science and Mathematics, which is why I consider those who raise it in this context to be either ill-informed or disingenuous.