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Both the negative results of gödel and the positive result of gentzen are given as consequences of technically precise formalisms
On page 82, a more or less hilbert-style formal system is given in three parts: first a set of propositional calculus axioms and one rule (modus ponens), then a set of predicate calculus axioms and two rules (universal introduction and existential elimination), and finally a set of non-logical axioms for a first-order language with
There are other books which also present the same negative and positive answers to the arithmetic completeness question, but this book by kleene presents a very thorough basis in propositional and predicate calculus along the way, for which metamathematical theorems are developed towards obtaining these answers