There is a common confusion as to what the word "completeness" means in mathematical logic. Let's take the specific example of the first-order language of arithmetic, with the Peano axioms PA (plus the usual logical axioms and rules of inference). By Goedel's completeness theorem, if S is true in every model of PA then there is a formal proof of S from PA. By Goedel's incompleteness theorem, there is a sentence S true in the standard model of PA (i.e., the natural numbers with the usual interpretation of the arithmetical symbols) but for which there is no formal proof from PA. So while S may be true in the standard model, there are other models of PA in which S is false. Another way to interpret the incompleteness of PA is to say that there is a sentence S so that neither S nor its negation is a theorem of PA (independence and consistency, in that order). Normally, when you're asking about completeness of a (possibly exotic) logical system--in which you have established a notion of formal logical inference--you're using the word in the sense of Goedel's completeness theorem: you have a class of models and a notion of what it means for a sentence to be true in a model, so that every theorem of the formal system is true in every model (soundness). It may be that your formal system is not strong enough to give you a proof of every true sentence, but when it is, you say the system is complete. Once you have a complete logical system, then you may ask as to whether a given collection A of axioms within that system is complete. One way for A to be incomplete is for there to be two models of A such that some sentence is true in one of the models and false in the other. The theory of linear orders, for example, is incomplete within what is known as classical first-order logic.