Gödel's incompleteness theorem states that PA is negation-incomplete. One theorem that has been shown to be independent of the axioms of PA is the theorem that it's impossible to lose the Hydra game.
The theorem is nonetheless viewed as true, and it is indeed provable in set theory. However, since PA is independent of this theorem assuming PA is consistent, we should be able to augment the axioms of PA with an axiom that would guarantee that the negation of the theorem is provable.
At this point some confusion arises. In the new formal system, the theorem is false, but nonetheless it's considered "intuitively" true. Have we just added an axiom that is in fact "intuitively" false to the system? Considering also that in PA there is a model where the statement is false, shouldn't the PA axioms too be viewed with some suspicion?