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Anatoly_Vorobey comments on [SEQ RERUN] The Dilemma: Science or Bayes? - Less Wrong
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Unfortunately, this turns out to be subtly wrong - see my update in a sibling comment.
That's a pity. Still, given any non-halting TM for which F cannot prove that, it is easy (costs very little additional constant complexity) to build a TM for which F also cannot prove that it does not return any x. And this still answers my original question (whether the threshold is very high) in the negative. So, bad for K-complexity, we need something better.