What’s your current opinion on whether Sawin’s proved statement that “any logically coherent, approximable probability distribution over statements in arithmetic which assigns probability 1 to true pi-1 statements will assign probability 0 to some true pi-2 statements” is a special case of Theorem 3.7 of Gaifman & Snir (1982)? Alex Mennen looked at it and thinks “probably not,” but I’m not in a position to tell.
I agree with “probably not” from my reading so far, but I don’t feel I’ve understood enough of the preceding sections in the paper to be confident. I would not be too surprised if some other theorem in that paper was relevant, either; I haven’t gotten through enough to say, except that it’s definitely exploring related questions.
I would not be too surprised if some other theorem in that paper was relevant, either; I haven’t gotten through enough to say, except that it’s definitely exploring related questions.
The statement Sawin proved refers to computable probability distributions over logical statements, and Gaifman & Snir only considered a class of probability distributions which are necessarily undefinable (by Tarski’s theorem, since their probability distributions are certain about the truth-values of all statements in L0, which is expressive enough to implement Peano Arithmetic). So I actually would be fairly surprised if the paper ended up containing something relevant to it.
What’s your current opinion on whether Sawin’s proved statement that “any logically coherent, approximable probability distribution over statements in arithmetic which assigns probability 1 to true pi-1 statements will assign probability 0 to some true pi-2 statements” is a special case of Theorem 3.7 of Gaifman & Snir (1982)? Alex Mennen looked at it and thinks “probably not,” but I’m not in a position to tell.
I agree with “probably not” from my reading so far, but I don’t feel I’ve understood enough of the preceding sections in the paper to be confident. I would not be too surprised if some other theorem in that paper was relevant, either; I haven’t gotten through enough to say, except that it’s definitely exploring related questions.
The statement Sawin proved refers to computable probability distributions over logical statements, and Gaifman & Snir only considered a class of probability distributions which are necessarily undefinable (by Tarski’s theorem, since their probability distributions are certain about the truth-values of all statements in L0, which is expressive enough to implement Peano Arithmetic). So I actually would be fairly surprised if the paper ended up containing something relevant to it.