idea
A system nobody can check at all
Then Gödel stops applying, because the theorem needs a system whose proofs a computer can check. Such a system can even be complete: flip a coin for every statement, “add it” or “add its opposite”, and you end up with a rulebook that settles everything.
But it would settle about half of the statements wrongly. It could be perfectly consistent and still “prove” false things about numbers. Complete is not the same as true.