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.
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