[ETA: oops, comments crossed; cousin_it originally posted a restatement of my followup question.]
Yes, that’s what I meant. Nit, though: I don’t think there’s any particular reason to think that [lim_m lim_n prior at (m,n)] will be equal to [lim_n lim_m prior at (m,n)].
I would hope that if the proposed limit exists at all, you could define it over a nice probability space: I imagine that the individual outcomes (“worlds”) would be all sets of sentences of the language of PA, the sigma algebra of events would be the same as for the probability space of a countable sequence of coin tosses (a standard construction), and the set of all complete theories extending PA would be measurable and have probability 1. (Or maybe that is too much to hope for...)
[ETA: oops, comments crossed; cousin_it originally posted a restatement of my followup question.]
Yes, that’s what I meant. Nit, though: I don’t think there’s any particular reason to think that [lim_m lim_n prior at (m,n)] will be equal to [lim_n lim_m prior at (m,n)].
I would hope that if the proposed limit exists at all, you could define it over a nice probability space: I imagine that the individual outcomes (“worlds”) would be all sets of sentences of the language of PA, the sigma algebra of events would be the same as for the probability space of a countable sequence of coin tosses (a standard construction), and the set of all complete theories extending PA would be measurable and have probability 1. (Or maybe that is too much to hope for...)