Roughly, Godel demonstrated that in a logical system, that is similar or as rigorous as mathematics, there are statements which maybe true, but are unprovable, within the system. If a statement is not provable an inconsistency or self-contradiction may or will develop that invalidates the system. Arrival at this point then demonstrates that a system has been considered or examined sufficently to move on. Should arrival at this point be the focus for examination of any system?

Roughly, Gödel demonstrated that in a logical system, that contains a model or arithmetic, there are statements which may be true, but are unprovable within the system. 

If a statement is not provable an inconsistency or self-contradiction may or will develop that invalidates the system. Arrival at this point then demonstrates that a system has been considered or examined sufficiently to move on. Should arrival at this point be the focus for examination of any system?