I would like to ask a question about Gödel's Incompleteness Theorems which I've had in the back of my head for some time. Since I'm a student working in a completely different area of maths (my usual pastime is cutting and pasting manifolds), my understanding of these results is nontechnical (I first learned about them before I started studying maths at university, by reading Nagel and Newman's excellent book).
As I understand it, the First Incompleteness Theorem is proven by producing a sentence, $G$ (which in English reads: "This sentence is not provable in the system."), that is neither provable nor refutable in the system in question. Nevertheless, $G$ is true, which means it satisfies the inductive definition of truth (right?). Here's what I don't understand:
Why does the sentence $G$ satisfy the inductive definition of truth? Is this because it is the negation of a false statement? If so, why is $\neg G$ false? Is this because it causes a contradiction, and sentences that lead to a contradiction in the system are false by definition?
Why doesn't the fact that $G$ is true (i.e. the list of steps reducing the truth of $G$ to the truth to atomic sentences (axioms?)) constitute a proof of $G$?
By Gödel's Completeness Theorem, there are formal systems in which $G$ is false, as argued here. How can this not cause a contradiction ($G$ can't both be provable and not be!). [What makes these systems different from those in which $G$ is true, and why is it often stated that $G$ is true, when in fact there are cases in which it isn't? Edit: answered here]
I hope I have managed to make myself clear. Thanks for any clarification!