Model existence theorem in set theory

Question

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?

Answer 1

You are asking for the completeness theorem of first-order logic, proved by Kurt Gödel in 1929.

There are various ways to state the completeness theorem, and among them are the following two assertions:

• Whenever a statement $\varphi$ is true in every model of a theory $T$, then it is derivable from $T$.

• Whenever a theory $T$ is consistent, then it has a model.

These assertions are easily seen to be equivalent, by the following argument. If the first holds, and a theory $T$ has no model, then false holds (vacuously) in every model of $T$, and so $T$ derives a contradiction; so the second holds. If the second holds, and $\varphi$ holds in every model of $T$, then $T+\neg\varphi$ has no models and so is inconsistent by 2, so by elementary logic, $T$ derives $\varphi$; so the first statement holds.

Answer 2

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.