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Godel's theorems mean something quite specific, and rely on an equally specific set of hypothesis.

It's tempting to try to apply them (or rather the same kind of conclusions) in other (non math) contexts, but it's very not obvious that you'll get something sensible. While you can play with the ideas, invoking Godel's theorem outside of its specific context doesn't make much sense.

I usually think of logics as search algorithms, since that's how the semantics for the meta-language describing them are defined.

I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.