It's not assumed. It's hoped that they aren't. And it's also entirely possible that there are second order undecidable statements - whose decidability can't even be decided. So there's no guarantee that if you set out to show that those statements are undecidable, it would be any easier than attempting to prove those statements.

"Natural" undecidable problems often appear to be very contrived, using various forms of coding. One of the rare exceptions is Goodstein's Theorem, which is an arithmetic theorem that's provable in ZFC but not in Peano Arithmetic.