A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel

A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.