“But Larry, PA does not actually say that 6 is prime, and 6 is not prime.”
Well of course 6 isn’t prime. But if PA said it was, then it would be. There’s nothing invalid about proving that A → B if you know ~A. It’s just not very useful. But for a somewhat less vacuous example, let RH be the riemann hypothesis. Then if PA |- RH then RH is true and if PA |- ~RH then RH is false. At least one of these implications has a false hypothesis, but they are both perfectly valid.
“But Larry, PA does not actually say that 6 is prime, and 6 is not prime.”
Well of course 6 isn’t prime. But if PA said it was, then it would be. There’s nothing invalid about proving that A → B if you know ~A. It’s just not very useful. But for a somewhat less vacuous example, let RH be the riemann hypothesis. Then if PA |- RH then RH is true and if PA |- ~RH then RH is false. At least one of these implications has a false hypothesis, but they are both perfectly valid.