An axiomatic system is complete if it can either prove or refute every possible statement. "Refute" means "prove the negation of". So a complete axiomatic syste

An axiomatic system is complete if it can either prove or refute every possible statement. "Refute" means "prove the negation of". So a complete axiomatic system is as powerful as possible w/o being inconsistent (if you prove both X and not-X, you're inconsistent). To say that a system is incomplete is to say that there are things it can neither prove nor refute. These things could be true or false, no matter. In fact, if X is one such "undecidable" statement, not-X is also one, and one of them has to be true, the other false. But it doesn't matter for the definition of "completeness". You could have axiomatic systems that are built on axioms some of which are known to be false, and yet they're consistent. These systems could be complete or incomplete. Think about "what's provable" and "what's true" as completely independent layers of information you can have about statements. It's very good when they partially coincide, for example, when all the things you can prove are in fact true things (which will happen if you start out from true axioms). But they don't have to. They could fail to coincide at all. Now for the layer of information "what's true" there is automatically a very nice property that either X or not-X is true, for any statement X. You get that for free just from the nature of what truth/falseness are. For the layer of information "what's provable" it's not at all guaranteed that either X or not-X is provable for any statement X (this is, again, what "completeness" means). If your axiomatic system is very weak, it can only prove a few paltry statements, and it will not nearly be complete, there'll be a huge universe of statements out there that it'll be too stupid to prove anything interesting about, either about X or about not-X. But you could hope that as you add more and more strong, powerful axioms to your system (and nevermind if they're true or not! it's intuitively easy to think of them as true, but they don't have to be), you get to be able to prove more and more stuff, and eventually you'll have a complete system. Goedel's result is the hugely surprising fact that you can't: once your system is powerful enough, and yet manageable (you can't just pour everything in as axioms) it'll have to be incomplete, because it's powerful enough to formalize statements about itself, and that kind of self-awareness will have to be (because of the reasons similar to the Liar's Paradox) an essential check on what it can prove.