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.
Lo! A cartoon proof of Löb's Theorem!
Löb's Theorem shows that a mathematical system cannot assert its own soundness without becoming inconsistent. Marcello and I wanted to be able to see the truth of Löb's Theorem at a glance, so we doodled it out in the form of a cartoon. (An inability to trust assertions made by a proof system isomorphic to yourself, may be an issue for self-modifying AIs.)
It was while learning mathematical logic that I first learned to rigorously distinguish between X, the truth of X, the quotation of X, a proof of X, and a proof that X's quotation was provable.
The cartoon guide follows as an embedded Scribd document after the jump, or you can download as a PDF file. Afterward I offer a medium-hard puzzle to test your skill at drawing logical distinctions.
Cartoon Guide to Löb's ... by on Scribd
Cartoon Guide to Löb's Theorem - Upload a Document to Scribd
And now for your medium-hard puzzle:
The Deduction Theorem (look it up) states that whenever assuming a hypothesis H enables us to prove a formula F in classical logic, then (H->F) is a theorem in classical logic.
Let ◻Z stand for the proposition "Z is provable". Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C.
Applying the Deduction Theorem to Löb's Theorem gives us, for all C:
However, those familiar with the logic of material implication will realize that:
Applied to the above, this yields (not ◻C)->C.
That is, all statements which lack proofs are true.
I cannot prove that 2 = 1.
Therefore 2 = 1.
Can you exactly pinpoint the flaw?