On wikipedia it is written that
Decidability should not be confused with completeness. For example, the theory of algebraically closed fields is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable.
A theory is called complete (see wikpedia:complete theory) if for every sentence either it or its negation is provable in the theory. But then, I guess completeness would yield decidability, as we can just enumerate all provable proposition (proofs are finite length derivations) and check whether the current one equals the sentence (or its negation) under question. By completeness this procedure will terminate.
So maybe completeness of the logical system is meant in that paragraph, i.e. a logical system is complete if the valid sentences coincide with the provable ones. By Gödel's completeness result first order logic is complete. As written here the theory of algebraically closed fields is axiomatizable in first order logic, so it could not be incomplete in this sense, but the cited paragraph claims exactly that.
So, for both interpretations of completeness, completeness of a theory, or of a logical system, the cited paragraph makes no sense to me. Could someone explain what I miss, or what is meant here?