Your description of “true arithmetic” seems to be unusually realist. Do you mean that, or is it meant to illustrate a way of thinking about what a logical probability on models is a ‘probability of’?
Pg. 5, 2^|−ψ| should be 2^-|ψ|.
I’m not sold on your discussion of desiderata for priors. Specifically, on the properties that a prior can have without its computable approximation having them—I almost feel like these should be split off into a separate section on desiderata for uncomputable priors over models. Because the probability distribution over models should update on deduction as the agent expends computational resources, I don’t think one can usefully talk about the properties of a (actually computed) prior alone in the limit of large resources. The entire algorithm for assigning logical probability is, I think, the thing whose limit should be checked for the desiderata for uncomputable priors.
Yes, the description of true arithmetic is a bit realist; I’m not particularly sold on the realism of true arithmetic, and yes, it’s mostly meant to illustrate a way of thinking about logical uncertainty. Basically, the question of logical uncertainty gets more interesting when you try to say “I have one particular model of PA in mind, but I can’t compute which one; what should my prior be?”
Typo fixed; thanks.
The entire algorithm for assigning logical probability is, I think, the thing whose limit should be checked for the desiderata for uncomputable priors.
I agree.
I don’t think one can usefully talk about the properties of a (actually computed) prior alone in the limit of large resources.
I see these desiderata more as a litmus test. I expect the approximation algorithm and the prior it’s approximating will have to be developed hand-in-hand; these desiderata provide good litmus tests for whether an idea is worth looking into. I tend to think about the problem by looking for desirable limits with approximation in mind, but I agree that you could also attack the problem from the other direction.
Your description of “true arithmetic” seems to be unusually realist. Do you mean that, or is it meant to illustrate a way of thinking about what a logical probability on models is a ‘probability of’?
Pg. 5, 2^|−ψ| should be 2^-|ψ|.
I’m not sold on your discussion of desiderata for priors. Specifically, on the properties that a prior can have without its computable approximation having them—I almost feel like these should be split off into a separate section on desiderata for uncomputable priors over models. Because the probability distribution over models should update on deduction as the agent expends computational resources, I don’t think one can usefully talk about the properties of a (actually computed) prior alone in the limit of large resources. The entire algorithm for assigning logical probability is, I think, the thing whose limit should be checked for the desiderata for uncomputable priors.
Yes, the description of true arithmetic is a bit realist; I’m not particularly sold on the realism of true arithmetic, and yes, it’s mostly meant to illustrate a way of thinking about logical uncertainty. Basically, the question of logical uncertainty gets more interesting when you try to say “I have one particular model of PA in mind, but I can’t compute which one; what should my prior be?”
Typo fixed; thanks.
I agree.
I see these desiderata more as a litmus test. I expect the approximation algorithm and the prior it’s approximating will have to be developed hand-in-hand; these desiderata provide good litmus tests for whether an idea is worth looking into. I tend to think about the problem by looking for desirable limits with approximation in mind, but I agree that you could also attack the problem from the other direction.