This notion, by the argument you give before the final theorem, implies belief (with probability 1) in all true Π1 sentences—therefore it must disbelieve true Π2 sentences. This is quite intriguing, but likely means it’s not quite what we want.
If uniformly coherent predictors exist, their self-referential knowledge would obey Paul Christiano’s reflection principle (which follows directly from believing true Π1 sentences).
It does not imply belief in all true Π1 sentences. If f(x) is true for all x, the probability of f(x) will go to 1 as x goes to infinity, but the probability of (¬¬)n∀xf(x) may stay bounded away from 1.
So the difference in this example is that a coherent prior with approximation Pt(n) can have limtPt(ϕsn)→1 for all n, but not have limtPt(ϕst)→1; uniformly coherent must have this. To be precise, if we set Pt(ϕ):=M(┌(¬¬)tϕ┐) with M uniformly coherent, we must have this.
This notion, by the argument you give before the final theorem, implies belief (with probability 1) in all true Π1 sentences—therefore it must disbelieve true Π2 sentences. This is quite intriguing, but likely means it’s not quite what we want.
If uniformly coherent predictors exist, their self-referential knowledge would obey Paul Christiano’s reflection principle (which follows directly from believing true Π1 sentences).
It does not imply belief in all true Π1 sentences. If f(x) is true for all x, the probability of f(x) will go to 1 as x goes to infinity, but the probability of (¬¬)n∀xf(x) may stay bounded away from 1.
Ahh, right. Silly mistake.
So the difference in this example is that a coherent prior with approximation Pt(n) can have limtPt(ϕsn)→1 for all n, but not have limtPt(ϕst)→1; uniformly coherent must have this. To be precise, if we set Pt(ϕ):=M(┌(¬¬)tϕ┐) with M uniformly coherent, we must have this.
Correct.