Well no, Gödel's incompleteness theorem doesn't really have anything to do with material implication. It tells us that there is some formula G such that neithe

Well no, Gödel's incompleteness theorem doesn't really have anything to do with material implication. It tells us that there is some formula G such that neither G nor ¬G is provable. Proof systems for relevance logic restrict valid derivations compared to classical propositional logic (relevance logic requires you to use the antecedent), so they can prove even fewer things.