If we think the speaker to be completely honest (incapable of saying false things), and also we have nonzero belief that the speaker will enumerate axioms of Peano Arithmetic, then we already believe a contradiction (because assuming that a proof of X implies X creates all the contradictions).
Adding to the axioms of PA, the statement “a proof of X from the axioms of PA implies X”, does not create any contradictions. This is just the belief that PA is sound.
What would be contradictory would be for PA itself to believe that PA is sound. It is fine for an agent to have the belief that PA is sound.
If we think the speaker to be completely honest (incapable of saying false things), and also we have nonzero belief that the speaker will enumerate axioms of Peano Arithmetic, then we already believe a contradiction (because assuming that a proof of X implies X creates all the contradictions).
Adding to the axioms of PA, the statement “a proof of X from the axioms of PA implies X”, does not create any contradictions. This is just the belief that PA is sound.
What would be contradictory would be for PA itself to believe that PA is sound. It is fine for an agent to have the belief that PA is sound.
There might be some technicality under which you’re not simply wrong. https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem