Note that Gödel's incompleteness theorems do not apply to just any system of logic: they are about particular formal systems that can prove certain facts about the arithmetics of integers. So, for them to fail, it doesn't even take a non-mathematical formal system, just something that has nothing to do with natural numbers, for example, Euclidean geometry, which happens to be fully decidable.
Godel's theorem is irrelevant to systems of concepts, conceptual analysis, or theorising in general. It's narrowly about technical issues in logic.
It has been "thematically appropriated" by a certain sort of pop-philosophy, but it says nothing relevant.
Philosophy isnt the activity of trying to construct logical embeddings in deductive proofs. If any one ever thought so, then there's some thin sort of relevance, but no one ever has.
