TITLE: Model existence theorem in set theory
QUESTION [5 upvotes]: From the FOM newsgroup I learned:

It's a theorem of (first-order) set
  theory that every consistent
  first-order theory has a model.

What's the exact formulation of this theorem in purely set-theoretic terms? (Reference?)
Is the following a sensible point of view?
Given a definition for "defining a consistent first-order theory" for formulas $\phi(x)$ in the language of set theory, including conditions that make $\phi(x)$ a "theory" and "consistent". Think of formulas $\phi(x)$ that say $x$ is a graph or $x$ is a group or $x$ is a topological space.
Can the model existence theorem then be seen as a theorem scheme such that for every formula $\phi(x)$ defining a consistent first-order theory (in the sense above) the sentence $(\exists x)\phi(x)$ is provable from the axioms of set theory?

REPLY [2 votes]: That's Gödel's Completeness theorem. The formulation in set theory is: "Every consistent first order theory has a model". An equivalent formulation is "Every logical consequence of a theory is provable from said theory".
This is not a scheme of theorems. It is a single theorem. Keep in mind though that this theorem doesn't speak about the metalanguage (in this case that of set theory), but rather about the sentences of various languages as seen inside set theory.The theorem now states that if a set of sentences is consistent, then there exists a set that is a model of these sentences. The standard definition of a model is a set equipped with various functions, relations etc. and a map between these and the objects of the language.