What technical books should I read to understand clearly Gödel's theorems and their implications for math?
I had a course in university which covered set theory, relations, functions, cardinality (with also Cantor's diagonal argument), boolean algebra viewed as a distributive lattice and Stone's theorem (without the proof), sentential calculus and first order logic (defined with syntax and semantic), tableaux method for both, Hilbert system and deduction theorem (and so, the sequent calculus). Also, without proving it, the professor said that there is a demonstration of soundness and completeness of the Hilbert system (all tautologies and only them can be derived).
After the course I read "Gödel's proof" by Nagel and Newmann and "The universal computer" by Martin Davis. Now I want to understand more. I want to understand how logic can embrace every mathematical reasoning, the difference between an axiomatic theory and a formal system and how a formal system can be built from an axiomatic theory (if it makes sense... maybe I only have confused ideas). And of course i would like to understand the real Gödel's incompleteness theorem proof without someone to simplify it for me (like in Nagel and Newmann's book).