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cousin_it comments on An example of self-fulfilling spurious proofs in UDT - Less Wrong Discussion

20 Post author: cousin_it 25 March 2012 11:47AM

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Comment author: cousin_it 26 March 2012 09:20:43AM *  1 point [-]

Here's another interesting followup question. My implementation of Q relies on enumerating all proofs in order, so running Q requires exponential time. Is there a "Henkin-style" implementation of Q that constructs the self-referential proof quickly and directly, maybe using the steps of Löb's theorem or something? That's how I first tried to construct Q after reading your comments, and failed, but it might still be possible.

Comment author: gRR 01 April 2012 07:37:40PM *  0 points [-]

Maybe something like this:

def Q():
return Proof1("IsValidProof(X, Q()) => X") +
Proof2("IsValidProof(X, Q())") +
X

where IsValidProof(Statement, Proof) is Goedel's Bew function, Proof1 is a formalization of the argument that "if Q() returns a valid proof of X, then A() will return 1, and X will be true", and Proof2 is a formalization of the argument that "a valid proof of A=>B, followed by a valid proof of A, followed by B, is a valid proof".

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