It is pretty incomprehensible. It relates to the nature of words like "proof" and just, without and doubt, those are tricky concepts. There's also a lot of imp

It is pretty incomprehensible. It relates to the nature of words like "proof" and just, without and doubt, those are tricky concepts. There's also a lot of implied background. It sounds simple to say what's going on in informal language and hope that it makes sense. That's not true, however. The statement of the theorem relies on the machinery of formal proof. That machinery and, more importantly, it's relation to day-to-day reasoning is big and complicated. --- A formal theorem is a statement in a language about the world. Typically speaking, these statements are not interesting without "quantification", the use of terms like "for all" or "there exists". Quantification makes formal theorems powerful by letting them speak about an entire universe of things all at once. The meaning of quantification is difficult. A sentence like "For all x, E(x)" is a discussion of the nature of the "universe" of x's. Sometimes we even refer to that explicitly when we say "For all x in X, E(x)" and upper-case X is that universe itself. For the theorem to be true, we need to not only pick an explicit universe X and then show that E(x) is true for any choice of x in X, but also we often have to show that this whole process continues to work no matter which universe X is chosen. This is the power of universality: making such an incredibly wide-reaching claim. (From a programming perspective this is the difference between types and tests. Tests show E(x) for all kinds of interesting x's. Types show "for all x in X, E(x)". There's some magic to this.) So one kind of way to show the validity of a theorem is to enumerate all universes (at least all of those that are considered "compatible" with the theorem) and then show that all choices of elements from that universe support the statement of the theorem. This is considered to be "semantic" meaning in that it's how theorems are practically useful. We only care about theorems when they've been instantiated in a practical way. Their value comes out of how freely we can choose to instantiate them. But there's another tricky way. Formal proofs are (usually) data structures constructed cleverly so that their shape relates to the syntactic structure of a theorem. They exploit the compositional nature of the theorem language and afford means to manipulate theorems (and their corresponding proofs) in a purely mechanical way. For instance, if I have theorems A and B as well as formal proofs of those theorems, a and b respectively, then I can have a proof of the composed theorem (A & B) by composing the sub-proofs together into a tuple like (a, b). This is a totally mechanical exploitation of the structure of (formal) proofs and theorems. The benefit of formal proofs is that syntactic manipulation can be verified very quickly. Given a theorem T and a (potential) proof P I can tell you whether P actually proves T very quickly. It's very difficult to find a proof P given only a theorem T. It's very difficult to write a theorem T which matches what you're interested in, but if you've got them all then I can verify your proof quickly. On the other hand, actually checking every universe where your theorem might hold and proving that it does is expensive no matter how you cut it. We'd like to be able to skip over that work. And that's what Gödel Completeness does. Gödel showed that when a theorem holds in all possible universes (it has semantic meaning) then there must be a formal proof of it (it has syntactic meaning). Since we're generally interested in semantic meaning (e.g. the applicability of a theorem to whichever universe(s) we happen to care about) then we're very excited to learn that syntactic meaning will mirror that semantic meaning. Without Completeness, syntactic proofs would still be valuable. A syntactic proof still implies semantic validity. What wouldn't be true is the applicability of syntactic reasoning. Maybe there are tons of important theorems out there which are true but can only be shown to be true through exhaustive enumeration? Suddenly mathematics as an act of human communication is fraught---many interesting things can only be explained exhaustively. There's often no way to "compress" meaning down into a pithy proof. But, no, completeness holds and formal proofs hold exactly the position of meaning we'd hope that they would. Well. At least for first-order theories.