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?
Best Answer
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.