Secondly, is there a reason why subjective Bayesian theory isn't mentioned? To me it seems obvious that expert elicitation and assigning probabilities to logical uncertainty is perfectly fine.
So, for example, fuzzy logic might help you quantify to what extent someone is "tall," where tallness admits of degrees rather than being binary. Or it might help you quantify to what extent a proof is "long." But it won't tell you how to calculate the subjective probability that there exists a proof of some theorem that is no more than 500 characters long in some fixed language. For that, you either need to find a proof, exhaustively demonstrate that no proof exists, or find a way to reason under logical uncertainty; and we haven't found any ways to use fuzzy logic to make progress on formalizing inference under logical uncertainty.
As for actually addressing logical uncertainty and asymptotic convergence, I think subjective Bayesianism can be used in both cases. For example you write "the axioms of probability theory force you to put probability either 0 or 1 on those statements", which I think is simply not true. If I as an "expert" claimed that "in my experience there is a 70% chance of conjecture being correct", I can set "Prior(conjecture)=0.7".
I like to hear you input on this.