The problem still continues. Even if you add the unprovable formula ( lets say formula1) in System1 as an axiom to System2 ( System1 + formula1 ), System2 will

The problem still continues. Even if you add the unprovable formula ( lets say formula1) in System1 as an axiom to System2 ( System1 + formula1 ), System2 will have its own unprovable formulas even though formula1 would be superficially provable in System2. You could add the unprovable formulas in System2 as axioms to System3 but System3 will have its own unprovable formulas . Godel showed that it's simply the nature of any formal system with complexity greater than or equal to arithmetic.