For simplicity we can make the “naive Bayesian” assumption that they’re all independent,
But then for that to work, your prior belief that x + 1 > x, for really large x, has to begin very close to 1. If there was some delta > 0 such that the prior beliefs were bounded above by 1 - delta then the infinite product would always be zero even after Bayesian updates.
How would you know to have a prior belief that reallybignumber + 1 > reallybignumber even in advance of noticing the universal generalization?
I cheat by assigning beliefs this way only in the “context” of checking this specific large statement :-) Maybe you could do it smarter by making the small statements non-independent. Will think.
ETA: yeah, sure. What we actually need is P(S1)*P(S2|S1)*P(S3|S1,S2)*… The convergence of this infinite product seems to be a much easier sell.
But then for that to work, your prior belief that x + 1 > x, for really large x, has to begin very close to 1. If there was some delta > 0 such that the prior beliefs were bounded above by 1 - delta then the infinite product would always be zero even after Bayesian updates.
How would you know to have a prior belief that reallybignumber + 1 > reallybignumber even in advance of noticing the universal generalization?
I cheat by assigning beliefs this way only in the “context” of checking this specific large statement :-) Maybe you could do it smarter by making the small statements non-independent. Will think.
ETA: yeah, sure. What we actually need is P(S1)*P(S2|S1)*P(S3|S1,S2)*… The convergence of this infinite product seems to be a much easier sell.