Question

Is there any reason to believe that big unsolved conjectures are either provable / disprovable in ZFC?

Answer 1

is it reasonable to assume they are provable/disprovable? Yes, because they certainly *seem* true and similar statements are actually provable/disprovable.

Is it possible they are independent  of ZFC? Of course.

Answer 2

I've heard that quite a few knowledgeable people think that P = NP is undecidable. That it it false is very unlikely, but it seems like (and I say that as an outsider) more people start to think that it isn't necessarily true either.

Answer 3

There's not a lot of reason to think the are or aren't independent.  


There is a compelling reason to think they won't be *proven* independent, which is that no one has any idea how to prove that these statements are independent that are a fraction as plausible as the ideas people have for how to prove them.