Regarding the issue :
completeness vs incompleteness
we have to understandt that the two notions are strictly connected (not only by the fact that Gödel was the author of both.
We have to take in account that Gödel's Completeness Theorem for First-Order Logic states :
if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally deducible (with FOL axioms and rules) by the axioms.
Gödel's Incompleteness Theorem is relative to formal system $F$ containing "a certain amount" of arithmetic (for example : Robinson Arithmetic, that is weaker than Peano's) and says that :
we can find a sentence $G_F$ expressible in that system, such that neither $G_F$ nor $\lnot G_F$ are deducible from the axioms.
This does not contradict the Completeness Theorem : being underivable from the axioms of $F$, the aforesaid sentences are not logical consequences of the axioms; this implies that they are not true in all models of the system $F$.
The proof of Gödel's Theorem shows us that the Gödel's sentence $G_F$ is true in the standard model; thus, it must be false in some non-standard model.
The arithmetical sentence originally constructed by Gödel in his proof is quite "un-natural", but starting form a result of Paris & Harrington (1977) has been possible, to find "natural" statements expressible in the language of arithmetic that are true but not provable in Peano Arithmetic.