I am new in mathematical logic and trying to understand the Godel's incompleteness theorem. If i'm not mistaken, it claims that in arithmetic there exists a statement $G$ such that there don't exist proofs of $G$ and $\sim G$ (if arithmetic and logic rules are consistent)
If I understand correctly, the key of the proof is to construct the expression $G$ which says "it doesn't exist a proof of $G$".
Help me to resolve the following paradox: From Godel's theorem follows that "it doesn't exist a proof of $G$". So it exists a proof that "it doesn't exist a proof of $G$", which is nothing but $G$. So it exists a proof of $G$
No, Gödel's theorem doesn't say "there is no proof of G." What it says is (as you yourself indicated in parentheses) "IF PA (Peano arithmetic) IS CONSISTENT, then there is no proof of G in PA." (That "IF" is actually an "IF AND ONLY IF" since, if PA is inconsistent, then everything including G is provable in PA.)
Now PA is more than strong enough to prove Gödel 's theorem. However, proving Gödel's theorem in PA does not prove that G is unprovable, it only proves that G is unprovable IF PA IS CONSISTENT.