Proofs are natural numbers via Gödel's encoding, or any other encoding scheme that you'd like (e.g. ASCII). The sentence "PA is inconsistent" can be stated in t

Proofs are natural numbers via Gödel's encoding, or any other encoding scheme that you'd like (e.g. ASCII). The sentence "PA is inconsistent" can be stated in the language of PA and it means "there is a number that encodes a valid proof from the axioms of PA to a contradiction (e.g. 0 ≠ 0)". This sentence is either true in all models of PA (in which case PA would be inconsistent, which we don't think it is), or it is true in some models and false in others - it can't be false in all models because otherwise, by completeness, there'd be a proof of PA's consistency in PA which is what the second incompleteness theorem rules out. The reason why this is not a category error is because of the self-referentiality of strong enough systems of logic.