axiomatizable

Axiomatizable describes a class of mathematical structures (or a theory) that can be completely described by a set of axioms in first-order logic. Concretely, a class K of structures in a fixed language is axiomatizable if there exists a set T of first-order sentences such that K is exactly the class of models of T (K = Mod(T)). In many discussions, the axiom set is required to be recursively enumerable, in which case the class is effectively axiomatizable.

Examples include the theory of groups, the theory of rings with unity, the theory of fields, and the theory of real closed fields; each can be captured by a (finite or infinite) set of first-order axioms. The class of finite groups, however, is not axiomatizable by any first-order theory; by the compactness theorem, no first-order theory can have exactly the finite groups as its models. More generally, finiteness properties are not first-order expressible.

Thus, a class is axiomatizable precisely when it is an elementary class: definable by a first-order theory. If the axioms are not effectively enumerable, the class is axiomatizable in the semantic sense but not effectively so. The notion is central in model theory because axiomatizability links structural descriptions with deductive systems and model-theoretic closure properties.