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cousin_it comments on Bounded versions of Gödel's and Löb's theorems - Less Wrong
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Yeah, if you want to make the bounds smaller, something like that might be the way to go. Paul Christiano described an explicit Loebian construction along these lines in an email on Apr 1, maybe you could repurpose that? I'm tired right now, so I guess I'll stick with the overkill bound :-)