- could not be both consistent and complete
- could not prove itself consistent without proving itself inconsistent
If a system cannot be complete and consistent what reasoning can we take help of, to design and implement a system that is useful. No need to worry! We have pearls in pile of human-knowledge-jewelry known as probability theory, theory of statistics at our disposal. These theories for example helped building randomized algorithms, learning theory which unquestionably have proven to be useful.
The research directions consequent to any negative result like incompleteness theorem are essentially examples of positive thinking. The incompleteness theorem implies conjectures which intuitively are true but hard to be proven are always possible thus ineffect guaranteeing room for growth or increased depth of understanding. Look out for the recent exuberance over the potentially complete proof of Poincare's conjecture by Grisha Perelman who purportedly denied the Fields medal.