Recommendation for a logic book to understand Godel's theorem

I have studied set theory but I couldn't understand even the first line of the Godel's proof.

For instance, $\omega^n$ means the set of functions from $\omega$ to $n$ in my set theory, ZFC,

but the proof I got says;

" For $R\subset \omega^n$ a relation, $\chi_R:\omega^n \rightarrow \omega$ is given by

$\chi_R (\overline{a}) = 1$ if $\neg R(\overline{a})$

$\chi_R (\overline{a}) = 0$ if $R(\overline{a})$ "

To me, it has to be $\omega \times ... \times \omega$ ($n$ times) for $R$ to be a relation, not $\omega^n$. What's going on?

Why does this use $\overline{a}$ rather than $a$, if it designates arbitrary one? Any reason?

I am not even sure whether function here has the same meaning as function defined in ZFC.

And does $R(\overline{a})$ mean "$\overline{a} \notin R$" ?

I don't understand why it says $\Delta$, a set of sentences of our language. Why is it set, not just collection?

That is, why we cannot form a collection of sentences consist of $\exists, \forall$ and so on, which is not a set?

There are several terminologies i don't understand too.

Computable, $\mu$ and so on.

Since these theorems are provable in any axiomatic system, i don't know what in ZFC is true and what in ZFC is not true in arbitrary axiomatic system. In other words, what is always true in any axiomatic system?

Please recommend me a nice book to study this. I don't want to go any further in mathematical logic than to understand these theorems.

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Actually, $\omega^n$ is the set of functions from $n$ to $\omega$, not the other way around. – Thomas Andrews Oct 16 '12 at 13:57
This question strikes me as very unfocused. The imcompleteness theorem, in particular, has lots of "popular" treatments out there of varying levels of detail. Forcing, on the other hand, is a deep tangle, a highly technical subject for which I have never seen an 'easy' introduction. The subject asks for references, the body asks specific questions about the text you are reading. – Thomas Andrews Oct 16 '12 at 14:01
@ThomasAndrews I'll change the title. Since those are only logic theorems i know, i typed it. – Rubertos Oct 16 '12 at 14:06
I found amazon.com/Incompleteness-Phenomenon-Martin-Goldstern/dp/… to be a very accessible introduction. I starts with basic logic, goes on to prove Gödel's Completeness Theorem, and then the Incompleteness Theorem. It also contains some model-theory (i.e., simply non-standard models of the integers), and also a proof of the model-theoretic version of the Completeness Theorem (the ultrafilter construction). – fgp Oct 16 '12 at 14:37