Definition:Axiomatization

Definition
Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

Let $\mathcal F$ be an $\mathcal L$-theory.

An axiomatization of $\mathcal F$ is a subset $\mathcal A \subseteq \mathcal F$ such that:


 * $\mathcal F = \left\{{\phi \in \mathcal L: \mathcal A \models_{\mathscr M} \phi}\right\}$

That is, all of $\mathcal F$ is a semantic consequence of $\mathcal A$.