[drchewbacca]

There is an interesting, if quite technical, answer here about Goodstein's Theorem which is a very reasonable number theory theorem which cannot be proven in first order logic.

So questions around statements which are true but not provable in certain logical systems do have concrete examples and are interesting imo.

https://math.stackexchange.com/questions/625223/do-we-know-i...

[zajio1am]

Goodstein's Theorem cannot be proved in Peano axiomatization of natural numbers in first order logic. You can have stronger axiomatization of natural numbers in first order logic that allows to prove Goodstein's Theorem - such axiomatization would contain subset of set theory and transfinite induction up to ordinal e_0. But it is still axiomatixation in first order logic.

[6gvONxR4sf7o]

As mentioned in that answer, these "natural independence phenomena" do seem much more interesting than Gödel's incompleteness results.