TITLE: How is the Gödel's Completeness Theorem not a tautology?
QUESTION [57 upvotes]: As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would become clearer for me, if I stated what I think the Gödel's Completeness Theorem is about (after studying various materials including Wikipedia it seems redundant for me) and someone knowledgeable would clarify my misconceptions. So here it goes:
As I understand, if we have a set $U$ with a particular structure (functions, relations etc.) we can interpret it (through a particular signature, e.g. group signature $\{ e,\cdot \}$ ), as a model $\mathfrak{A}$ for a certain mathematical theory $\mathcal{T}$ (a theory being a set of axioms and its consequences). The theory is satisfied by $\mathfrak{A}$ only if $U$'s structure satisfies the axioms.
Enter Gödel's theorem: For every first order theory $\mathcal{T}$ :
$$\left( \exists \textrm{model } \mathfrak{A}:
\mathfrak{A} \models \mathcal{T} \right) \iff \mathcal{T} \textrm{ is consistent}$$
So I'm confused. Isn't $\mathcal{T}$ being consistent a natural requirement which implicates that a set $U$ with a corresponding structure  always exists (because of the ZFC's set theory freedom in constructing sets as we please without any concerns regarding what constitutes the set)? And that in turn always allows us to create a model $\mathfrak{A}$ with an interpretation of the signature of the theory $\mathcal{T}$ in terms of $U$'s structure?
Where am I making mistakes? What concepts do I need to understand better in order to be able to properly comprehend this theorem and what model theory is and is not about? Please help!

REPLY [3 votes]: I want to suggest you some additional reflections on this topic.
You can see this post for an "historical" overview, aimed to show that the current "mainstream" reading of Gödel's Completeness Theorem as :


for f-o theory, satisfiability is equivalent to consistency


was not the "driver" to Gödel's discovery.
The original context was :

the 2nd of famous Hilbert's problems regarding the consistency of arithmetic

and the "call for a solution" contained into the first mathematical logic textbook : David Hilbert & Wilhelm Ackermann, Principles of Mathematical Logic (1st german ed - 1928).
In the second edition (1938) [citation from the english translation (1950), page 95] :

we may ask whether we have completeness in the other sense [see page 42 : The completeness of an axiom system may be defined in two ways. First, it may be taken to mean that all the true formulas of a certain domain which is characterized by content can be proved from 
  the set of axioms.]  The question here is whether all universally valid formulas of the predicate calculus, as defined at the beginning of this chapter, can be proved in the axiom system. We actually do have completeness in this sense. The proof is due to K.Gödel [Die Vollstandigkeit der Axiome des logischen Funktionen-kalkuls, Mh.Math.Physik, Vol.37 (1930).]

Recall also the comment [page 88] about the (proven) consistency of the axiomatization of f-o calculus :

We must not, by the way, overestimate the significance of this consistency proof [of f-o logic]. It amounts to saying that we assume the domain of individuals underlying the axioms to consist of only a single element, and thus to be finite. We have absolutely no assurance that the formal introduction of postulates unobjectionable as regards content leaves the system of theorems consistent.
For example, the question remains unanswered whether the addition of mathematical axioms would not, in our calculus, make any arbitrary formula provable. This problem, whose solution is of fundamental importance for mathematics, is incomparably more difficult than the question dealt with here. The mathematical axioms actually assume an infinite domain of individuals, and there are connected with the concept of infinity the difficulties and paradoxes which play a role in the discussion of the foundations of mathematics. 

Now we have on the scene the three "actors" of the play :

the adequacy of axioms and rules of "restricted predicate calculus" (i.e.f-o logic) regarding the notion of (universally) valid formula (solved in the positive by Gödel's Completeness Th)
the paramount relevance of consistency proofs for "fundamental" mathematical theories, like arithmetic and set theory (see 2nd Hilbert's problem), as a "solution" to the foundational problem (Frege, Russell's paradox, Intuitionism)
the search for a proof of "another kind" of completeness. 

In this context, we can read again the above passage from H&A :


The completeness of an axiom system [...] may be taken to mean that all the true formulas of a certain domain which is characterized by content can be proved from the set of axioms


as regarding also arithmetic. In this case, it amount to the question if [first-order version of] Peano's axioms are sufficient [or adequate, i.e.complete] to prove all arithmetical truth expressible in f-o language.
And again Gödel gave us the answer : NO, with his Incompleteness Theorems.