Hm; we could add an uninterpreted predicate symbol Q(n) to the language of arithmetic, and let s≡Q(0) and rn≡Q(¯¯¯¯¯¯¯¯¯¯¯¯¯n+1). Then, it seems like the only barrier to recursive enumerability of T∞ is that P’s opinions about Q(⋅) aren’t computable; this seems worrying in practice, since it seems certain that we would like logical uncertainty to be able to reason about the values of computations that use more resources than we use to compute our own probability estimates. But on the other hand, all of this makes this sound like an issue of self-reference, which is its own can of worms (once we have a computable process assigning probabilities to the value of computations, we can consider the sentence saying “I’m assigned probability <12” etc.).
Hm; we could add an uninterpreted predicate symbol Q(n) to the language of arithmetic, and let s≡Q(0) and rn≡Q(¯¯¯¯¯¯¯¯¯¯¯¯¯n+1). Then, it seems like the only barrier to recursive enumerability of T∞ is that P’s opinions about Q(⋅) aren’t computable; this seems worrying in practice, since it seems certain that we would like logical uncertainty to be able to reason about the values of computations that use more resources than we use to compute our own probability estimates. But on the other hand, all of this makes this sound like an issue of self-reference, which is its own can of worms (once we have a computable process assigning probabilities to the value of computations, we can consider the sentence saying “I’m assigned probability <12” etc.).