There are examples of "natural" sentences which are unprovable in Peano Arithmetic (which is one of the simpler systems for which Gödel's theorems apply): https://en.m.wikipedia.org/wiki/Paris%E2%80%93Harrington_the...

But in general, Gödel applies to all formal systems that satisfy certain properties, it's just that the exact unprovable sentences will be different (since you may just add that sentence as an axiom) that's why the general example is very abstract. It shows that sufficiently rich theories are not only incomplete, but also incompletable.