Sorry for the naive question. Is it theoretically possible that
a statement such as "there are infinitely many primes of the form n^2+1"
could be true, but not provable, in ZFC?

I'm well aware that there are going to be statements in mathematics which
are true but not provable, but on the other hand I also am half-aware that
it's too naive to expect that an *arbitrary* statement in mathematics, whose
truth value is unknown, might be true but not provable. As a concrete
example, I have a vague memory that, before Fermat's Last Theorem
was resolved, someone told me that the statement
"Fermat's Last Theorem is wrong" could not possibly be true but not
provable in ZFC. My knowledge of proof theory is sufficiently vague
for me not to be able to back this up rigorously, but the force
of the idea was that if FLT really were false then there would
exist a counterexample, and "writing down the counterexample" would be
a proof that FLT was false. But I am sufficiently confused about
these sorts of ideas to understand how far they can take you, and,
in particular, whether they go as far as to apply to the above question
about primes (I suspect they don't...).

Kevin Buzzard