Earlier quoted context omitted.
That isn't quite right. With a sufficiently powerful formal system, you're forced to either have inconsistency or incompleteness - you're describing a system that is inconsistent. It's usually much better to have consistency and to sacrifice completeness. Then you'll have Ps that are true but unprovable, but at least you won't have P=~P which makes the system rather useless.
What does true but unprovable mean? What happens if you take such a proposition, negate it and add as an axiom?
That being said, here is a much better resource than I am: http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_t...