Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.
The theorem is only true for the phi that it’s true for.
The claim that phi must be true because if it’s true then it’s true, and if it’s false then “if PA |- phi then phi” has an officially true conclusion whenever PA does not imply phi, is bogus.
It’s simply and obviously bogus, and I don’t understand why there was any difficulty about seeing it.
Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.
The theorem is only true for the phi that it’s true for.
The claim that phi must be true because if it’s true then it’s true, and if it’s false then “if PA |- phi then phi” has an officially true conclusion whenever PA does not imply phi, is bogus.
It’s simply and obviously bogus, and I don’t understand why there was any difficulty about seeing it.