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shminux comments on Reflection in Probabilistic Logic - Less Wrong
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Another stupid question: would relaxing the schema 3 to incomplete certainty allow the distribution P to be computable? If so, can you get arbitrarily close to certainty and still keep P nice enough to be useful? Does the question even make sense?
The computability problem is due to the unbounded ability of P to reason about theorems in the base theory (so that would be a necessary point of relaxation). A computable total function can't even assign values > 1/2 to all the theorems of PA and values < 1/2 to all the sentences refuted by PA.