Back in July 2013, Will Sawin (Princeton) and Abram Demski (USC) wrote a technical report describing a result from that month’s MIRI research workshop. We are finally releasing that report today. It is titled “Computable probability distributions which converge on believing true Π1 sentences will disbelieve true Π2 sentences.”
It might seem reasonable that after seeing unboundedly many examples of a true Π1 statement that a rational agent ought to be able to become increasingly confident, converging toward probability 1, that this statement is true. However, we have proven that this plus some plausible coherence properties, necessarily implies arbitrarily low limiting probabilities assigned to some short true Π2 statements.