Axiomatizability results express a rather subtle relationship between semantics and syntax. They give complete characterizations of certain classes of theories in purely semantic terms, formulated as closure properties of classes of models under some categorical operators. Perhaps the most famous example is the Birkhoff Variety Theorem of equational logic: a class of algebras for a signature is closed under products, sub-algebras, and homomorphic images (i.e. quotients) if and only if it is the class of algebras of an equational theory. This result is considered as one of the developments that represented the dawn of model theory.