Can something be both true and unprovable without being a self-referential Godel sentence? I assume most statements that aren't self-referential are independent, like the continuum hypothesis with ZFC.
Godel sentences are only true if you assume consistency. And once you add that axiom, the computer can prove them too.
G is independent of your original system T (cannot be proven true or false).
If you create T+ with added consistency axiom, then G is true and provable.
But G is never true and unprovable in the same system.
It's a lot like is/ought or theology.