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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 from yudkowsky.net as a PDF file. Afterward I offer a medium-hard puzzle to test your skill at drawing logical distinctions.
Read this document on Scribd: Cartoon Guide to Löb's Theorem
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?