The way I see it is that both theorems are statements about the relationship between a theory and models of that theory.
Goedel's completeness theorem tells you that for a theory in first-order logic, there's a model that corresponds exactly to that theory: anything that's true in that model can be proven in the theory, and vice versa.
Godel's incompleteness theorem tells you that there's no such model for any system that includes the usual "full" version of arithmetic. A model is a very concrete thing, so any question you ask about the model has an answer - but you don't necessarily know whether that answer is a fact about the theory (i.e. true in all models of the theory - which any statement that's provable would be) or just a fact about that particular model.
It's like if you write some C code, run it, and it prints 4. Does that mean your program always prints 4 according to the C standard? Or did your program do some undefined behaviour, and it just happened that with your particular compiler it printed 4?
> there's a model that corresponds exactly to that theory: anything that's true in that model can be proven in the theory, and vice versa.
Unfortunately it doesn't work if you swap our "for all models" with "there exists a model."
Take the theory of the natural numbers with the new symbol "Special" and the axiom "there exists a number n such that Special(n) holds." In every model of that theory there will be a concrete n that is Special (say 6). However, the theory will never be able to prove that a specific n is Special, precisely because it is under-specified which n it is.
> Godel's incompleteness theorem tells you that there's no such model for any system that includes the usual "full" version of arithmetic.
No in conjunction with Godel's completeness theorem, the incompleteness theorem would say rather that there is no theory that is strictly only the "full" version of arithmetic. Any theory will always be too broad and have too many models that satisfy it, only one of which is the "usual" model of arithmetic.
EDIT: Ah I see I misinterpreted your statement the first time, around, but my objections still hold with modification. Change "the theory of the natural numbers" to a far weaker theory, e.g. the theory of Presburger Arithmetic, and everything else still holds.
> Take the theory of the natural numbers with the new symbol "Special" and the axiom "there exists a number n such that Special(n) holds." In every model of that theory there will be a concrete n that is Special (say 6). However, the theory will never be able to prove that a specific n is Special, precisely because it is under-specified which n it is.
Hmm. So which n is special in the model that you get from the model existence theorem? Doesn't it all get somehow equivalence-classed away?
> No in conjunction with Godel's completeness theorem, the incompleteness theorem would say rather that there is no theory that is strictly only the "full" version of arithmetic. Any theory will always be too broad and have too many models that satisfy it, only one of which is the "usual" model of arithmetic.
Sure. The point is that there's always a gap between the theory and the model, an "implementation-defined" part of the model.
> Hmm. So which n is special in the model that you get from the model existence theorem? Doesn't it all get somehow equivalence-classed away?
The model existence theorem relies on extending your original language (and therefore your theory) with a bunch of new constants. So in this case you would end up conjuring up a specific new constant symbol just for `Special`.
You then "forget" about all those constants when you retract back to your original theory and so you lose the one-to-one relationship.
So it is true that if your original theory is Henkin (i.e. there is a constant symbol witnessing every single existential statement you have) and maximally consistent, then there exists a model that coincides exactly with your theory, in particular the model that is constructed from all your constant symbols modulo equality in your theory.
But most theories are not Henkin. And moreover most theories are not maximally consistent (since that would imply completeness in the sense of Godel's incompleteness theorems). In particular the theory you get out of the model existence theorem is usually not computably enumerable, so you can't actually write down your extended theory.
So most theories do not have that nice property that they correspond exactly with the semantic properties of a model.
Just want to point out that your answer is essentially incorrect. Godel's completeness theorem has nothing to do with the existence of models. It is about the provability of sentences that are true within all models.
Also, the incompleteness theorem doesn't say anything about the nonexistence of a model. It gives the existence of at least one sentence X (for sufficiently nice theories that include enough arithmetic) such that there ARE two models where X is true in one and false in the other.
> Godel's completeness theorem has nothing to do with the existence of models.
I wouldn't say that it has "nothing to do" with existence of models. It wasn't the original Goedel's formulation of the theorem, but these days, one of the most, if not the most popular statement of it is to say that "if theory T is consistent, there exists a model of it".
Goedel's completeness theorem tells you that for a theory in first-order logic, there's a model that corresponds exactly to that theory: anything that's true in that model can be proven in the theory, and vice versa.
Godel's incompleteness theorem tells you that there's no such model for any system that includes the usual "full" version of arithmetic. A model is a very concrete thing, so any question you ask about the model has an answer - but you don't necessarily know whether that answer is a fact about the theory (i.e. true in all models of the theory - which any statement that's provable would be) or just a fact about that particular model.
It's like if you write some C code, run it, and it prints 4. Does that mean your program always prints 4 according to the C standard? Or did your program do some undefined behaviour, and it just happened that with your particular compiler it printed 4?