“Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?”
I’m not stating that proving implications with false antecedent is particularly useful, just that it is valid. Also aside from 6 being prime it is true that for any sentence phi, ZF |- “if PA |- phi then phi”, but that ZF cannot even say, yet alone prove that “forall phi. if PA |- phi then phi”. But it can prove “forall phi. if PA |- phi then N |= phi”.
J Thomas: “How is that useful?”
I’m just answering your question
“Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?”
I’m not stating that proving implications with false antecedent is particularly useful, just that it is valid. Also aside from 6 being prime it is true that for any sentence phi, ZF |- “if PA |- phi then phi”, but that ZF cannot even say, yet alone prove that “forall phi. if PA |- phi then phi”. But it can prove “forall phi. if PA |- phi then N |= phi”.