Consider a finitely axiomatized theory $T$ with axioms $\phi_1,...,\phi_n$ over a first-order language with relation symbols $R_1,...,R_k$ of arities $\alpha_1,...,\alpha_k$. Consider the atomic formulas written in the form $(x_1,...,x_{\alpha_j})\ \varepsilon R_j$.
Translate this theory into a (finite) set-theoretic definition
T(X) :\equiv (\exists R_1)...(\exists R_k) R_i \subseteq X^{\alpha_i} \wedge \phi'_1 \wedge ... \wedge \phi'_n
where \phi'_i is $\phi_i$ with $(\forall x)$ replaced by $(\forall x \in X)$ and $(x_1,...,x_{\alpha_j})\ \varepsilon R_j$ replaced by $(x_1,...,x_{\alpha_j})\ \in R_j$ with $(x_1,...,x_{\alpha_j})$ an abbreviation for ordered tuples.
To show that $T$ has a model — i.e. to show that $T$ is consistent — is to prove the statement $(\exists x) T(x)$ from the axioms of set theory.
It is essential that the relations fulfill the conditions $\phi_i$ simultaneously. Thus it is not clear at first sight, how the existence of a model of a theory can be proved (or even be stated set-theoretically) that is not finitely axiomatizable, since it cannot be translated into a finite sentence.
Some other things are not clear (to me):
In this setting, doesn't the consistency of every theory dependend on the consistency of the choosen set theory? (If the set theory isn't consistent, every theory has a model.)
Furthermore, doesn't the consistency of a theory depend on the choice of the set theory in which $(\exists x) T(x)$ is proved? (In some set theories $(\exists x) T(x)$ can be proved, in others maybe not.)
What conditions has a theory to fulfill to be able to play the role of set theory in this setting? [It doesn't have to be the element relation $\in$ which $\varepsilon$ is mapped on. But one needs to be able to build ordered tuples of arbitrary length. What else? Something like powersets (since $R_i \subseteq X^{\alpha_i}$ is $R_i \in \mathcal{P}(X^{\alpha_i})$)? Is extensionality necessary? What is the general framework to discuss such questions?]