But Larry, PA does not actually say that 6 is prime, and 6 is not prime.
You could say that if PA proved that every theorem is false then every theorem would be false.
Or what would it mean if PA proved that Lob’s theorem was false?
It’s customary to say that any conclusion from a false premise is true. If 6 is prime then God’s in his heaven, everything’s right with the world and we are all muppets. Also God’s in hell, everything’s wrong with the world, and we are all mutant ninja turtles. It doesn’t really matter what conclusions you draw from a false premise because the premnise is false.
Your argument about what conclusion we could draw if PA said that 6 is prime is entirely based on a false premise. PA does not say that 6 is prime.
But Larry, PA does not actually say that 6 is prime, and 6 is not prime.
You could say that if PA proved that every theorem is false then every theorem would be false.
Or what would it mean if PA proved that Lob’s theorem was false?
It’s customary to say that any conclusion from a false premise is true. If 6 is prime then God’s in his heaven, everything’s right with the world and we are all muppets. Also God’s in hell, everything’s wrong with the world, and we are all mutant ninja turtles. It doesn’t really matter what conclusions you draw from a false premise because the premnise is false.
Your argument about what conclusion we could draw if PA said that 6 is prime is entirely based on a false premise. PA does not say that 6 is prime.