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:
((◻C)->C)->C
However, those familiar with the logic of material implication will realize that:
(X->Y)->Y
implies
(not X)->Y
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?
simon: Let me explain some of the terminology here, because that may be where the confusion lies.
A scentence is a finite string symbols that satisfies a certain set of syntactic constraints.
A theory is a (possibly infinite) set of sentences. PA is a theory.
A proof from a theory T is a finite set of sentences, each of which is either an element of T, or follows from the ones before according to a fixed set of rules.
The notation PA + X, where X is a sentences is just the union of PA and {X}.
The notation PA |- Y means that a proof from PA that ends in Y exists.
Now I have left out some technical details, like what exactly are the syntactic constraints on sentences, and what is the fixed set of rules for proofs, but we have enough to say what the deduction theorem means. It says
PA + X |- Y => PA |- "X -> Y"
or in english: if there is a proof from the theory PA + X to the scentence Y, then there is a proof from PA alone that X->Y.
So, you see, the deduction theorem is really just a technical lemma. It meerly shows that (in one particular way) our technical definition of a first order proof behaves the way it ought to.
Now on to Lob's theorem, which says that if PA |- "◻C -> C" then PA |- "C". Now in general if you want to prove PA |- A implies that PA |- B, one way to do it is to write a first order proof inside of PA that starts with A and ends with B. But that is not what is going on here. Instead we start with a proof of "◻C->C" and do something totally different than a first order proof inside of PA in order to come up with a proof that PA |- C.