This prior isn’t trollable in the original sense, but it is trollable in a weaker sense that still strikes me as important. Since μ must sum to 1, only finitely many sentences S can have μ(S)>ϵ for a given ϵ>0. So we can choose some finite set of “important sentences” and control their oscillations in a practical sense, but if there’s any ϵ>0 such that we think oscillations across the range (ϵ,1−ϵ) are a bad thing, all but finitely many sentences can exhibit this bad behavior.
It seems especially bad that we can only prevent “up-to-ϵ trolling” for finite sets of sentences, since in PA (or whatever) there are plenty of countable sets of sentences that seem “essentially the same” (like the ones you get from an induction argument), and it feels very unnatural to choose finite subsets of these and distinguish them from the others, even (or especially?) if we pretend we have no prior knowledge beyond the axioms.
This prior isn’t trollable in the original sense, but it is trollable in a weaker sense that still strikes me as important. Since μ must sum to 1, only finitely many sentences S can have μ(S)>ϵ for a given ϵ>0. So we can choose some finite set of “important sentences” and control their oscillations in a practical sense, but if there’s any ϵ>0 such that we think oscillations across the range (ϵ, 1−ϵ) are a bad thing, all but finitely many sentences can exhibit this bad behavior.
It seems especially bad that we can only prevent “up-to-ϵ trolling” for finite sets of sentences, since in PA (or whatever) there are plenty of countable sets of sentences that seem “essentially the same” (like the ones you get from an induction argument), and it feels very unnatural to choose finite subsets of these and distinguish them from the others, even (or especially?) if we pretend we have no prior knowledge beyond the axioms.