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AlexMennen comments on Probabilistic Löb theorem - Less Wrong Discussion
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Maybe this would be obvious if I knew anything about logic, but how do you know the system is consistent?
We don't - generally we build systems where we can show "system X is consistent iff Peano Arithmetic is consistent". And we assume that PA is consistent (or we panic).
Sorry my phrasing was bad; I actually do know that much about logic. But how do you know that this system is consistent iff Peano Arithmetic is consistent?
We don't have that system yet! Just that that is what we generally do with the systems we have.
The result linked at the beginning shows that there exists, in principle, a coherent probability distribution with certain properties. Edit: in particular, it assigns probability 0 to F or any other contradiction. And while it doesn't always (ever?) know the exact probability it assigns, it does know that P(F)<1-a for any a<1. That statement itself has probability 1. Therefore the part about violating the probabilistic Lob's Theorem clearly holds.
I can't tell at a glance if the distribution satisfies derivation principle #3, but it certainly satisfies #1.