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?