I'm currently studying Gödel's Incompleteness Theorem and I am in doubt about his use of primitive recursive functions.
To study it, i'm in the point of view of a system of predicate logic with the theory of Peano Arithmetic. That predicate logic has the extensions of equality (symbol $=$ defined) and function symbols.
1) In Gödel's proof he defines in some point the primitive recursive functions. Those definitions are necessary even if i'm using the function symbols extensions for predicate logic? The function symbols already offer an way of composing functions. An function $f \in F$ has the form of $f(t_1, t_2, \dotsc, t_n)$ where $t_n$ is any term of the logical system. But a function $f$ represents a term too. So we can compose functions like in $f(g(x))$ where $t_1=g(x)$. I don't see where the primitive recursion rule would give some benefit.
2) Another doubt is whether the primitive recursive functions is in the syntax domain or in the semantics domain. If in the semantic domain, how can I use expressions from inside the logical system? For example, in defining those functions i use the concepts $\forall$, $\exists$, $\land$ and the like.
3) At some point in the proof, he introduces the representability concept. It afirms that any definable relation can be expressed as a formula of the system. Why is this used for?
Thank you in advance.