There's been some prior discussion here about the problem of uncertainty of mathematical statements. Since most standard priors (e.g. Solomonoff) assume that one can do a large amount of unbounded arithmetic, issues of assigning confidence to say 53 being prime are difficult, as are issues connected to open mathematical problems (e.g. how does one discuss how likely one is to estimate that the Riemann hypothesis is true in ZFC?). The problem of bounded rationality here seems serious.

I've run across something that may be related, and at minimum seems hard to formalize. For a mathematical statement A, let F(A) be "A is proveable in ZFC) (you could use some other axiomatic system but this seems fine for now). Let G(A) be "A will be proven in ZFC by 2050". Then one can give examples of statements A and B, where it seems like P(F(A)) is larger than P(F(B)) but the reverse holds for P(G(A)) and P(G(B)).

The example that originally came to mind is technical: let A be the statement "ZPP is contained in P^X where X is an oracle for graph isomorphism" and let B be the statement "ZPP is contained in P^Y where Y is an oracle that answers whether Ackermann(n)+1 has an even or odd number of distinct prime factors." The intuition here is that one expects Ackermann(n)+1 to be essentially random in the parity of its number of distinct prime factors, and a strong source of pseudorandom bits forces collapse of ZPP. However, actually proving that Ackermann(n)+1 acts this way looks completely intractable. In contrast, there's no strong prior reason to think graph isomorphism has anything to do with making ZPP type problems easier (aside from some very minor aspects) but there's a lot of machinery out there that involves graph isomorphism and people thinking about it.

So, is this sort of thing meaningful? And are there other more straightforward, less complicated or less technical examples? I do have an analog involving not math but space exploration. P(Life on Mars) might be lower than P(Life on Europa) even though P(We discover life on Mars in the next 20 years) might be higher than P(We discover life on Europa in the next 20 years) simply because we send so many more probes to Mars. Is this a helpful analog or is it completely different?