A class K of structures of a signature σ is called an elementary class if there is a first-order theory T of signature σ, such that K consists of all models of T, i.e., of all σ-structures that satisfy T. If T can be chosen as a theory consisting of a single first-order sentence, then K is called a basic elementary class.
More generally, K is a pseudo-elementary class if there is a first-order theory T of a signature that extends σ, such that K consists of all σ-structures that are reducts to σ of models of T. In other words, a class K of σ-structures is pseudo-elementary if and only if there is an elementary class K' such that K consists of precisely the reducts to σ of the structures in K'.
For obvious reasons, elementary classes are also called axiomatizable in first-order logic, and basic elementary classes are called finitely axiomatizable in first-order logic.
These definitions extend to other logics in the obvious way, but since the first-order case is by far the most important, axiomatizable implicitly refers to this case when no other logic is specified.
The signatures that are considered in general model theory are often infinite, while a single first-order sentence contains only finitely many symbols.
Therefore, basic elementary classes are atypical in infinite model theory.
Hence, elementary classes are not very interesting for finite model theorists.
Let σ be a signature consisting only of a unary function symbol f. The class K of σ-structures in which f is one-to-one is a basic elementary class.
This is witnessed by the theory T, which consists only of the single sentence Let σ be an arbitrary signature.
The infinite σ-structures are precisely the models of the theory But K is not a basic elementary class.
Otherwise the infinite σ-structures would be precisely those that satisfy a certain first-order sentence τ.
By the compactness theorem, for some natural number n the set
But this is absurd, because this theory is satisfied by any finite σ-structure with
{f}, where f is a unary function symbol, such that K consists exactly of the reducts to σ of σ'-structures in K'.
Finally, consider the signature σ consisting of a single unary relation symbol P. Every σ-structure is partitioned into two subsets: Those elements for which P holds, and the rest.
Let K be the class of all σ-structures for which these two subsets have the same cardinality, i.e., there is a bijection between them.
This class is not elementary, because a σ-structure in which both the set of realisations of P and its complement are countably infinite satisfies precisely the same first-order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable.
Since this is also true for every signature extending σ, K is not even a pseudo-elementary class.
This example demonstrates the limits of expressive power inherent in first-order logic as opposed to the far more expressive second-order logic.