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Giles comments on Gödel and Bayes: quick question - Less Wrong
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Gödel's First Incompleteness Theorem involves statements of the form "this statement cannot be proved" rather than "this statement isn't true". (Mathematical truth is a surprisingly slippery concept to define).
So I think the analogous statement would be "The inferred probability of this statement is less than .5 given these priors and this evidence".