Everyone is always despairing about Godel's theorems but they're actually important and good: tehy're what makes science possible. To do science experiments you need a rigorous language for defining exactly hwat you're looking for, but you need propositions that aren't mere tautologies.
The need to introduce new axioms in order to syntactically arrive at certain propositions is what allows one to shape the evolution of a theory by creating an empirical basis for whther or not a syntactically unreachable proposition is "true" or not.
Experiments are used to settle those undecidable propositions, which allows them to have a rigorous definition in relation to the paradigm while directly shaping theoretical developments instead of just being trivially coextensive with what the theory implies