A class of structures that is exactly the class of models of some first‑order theory, equivalently a class closed under elementary equivalence and under taking elementary substructures.

Principle

Elementary classes are those definable at the level of first‑order logic: membership can be characterized by satisfaction of a set of first‑order sentences and is stable under the preservation notions specific to first‑order semantics (elementary substructures and elementary extensions).

Demonstration

The class of algebraically closed fields of characteristic 0 is elementary because it is the class of models of the first‑order theory ACF_0; any elementary substructure of an algebraically closed field of characteristic 0 is again algebraically closed of characteristic 0.

Misapplication

Calling a class elementary because it is closed under isomorphism alone is insufficient; for example, the class of finite groups is closed under isomorphism but is not elementary because finiteness cannot be expressed by first‑order sentences.

Consequence

If a class is elementary, model‑theoretic tools apply: one can consider complete theories, prime and saturated models inside the class, and exploit compactness and Löwenheim‑Skolem results to analyze the spectrum and definable sets in members of the class.

Reversal

A non‑elementary class (for instance an abstract elementary class or a class definable only in an infinitary logic) may still be robust but lacks a characterisation by a first‑order theory and may not be closed under elementary substructures or elementary equivalence.

Boundary

The notion excludes classes defined only by second‑order or infinitary conditions and those requiring cardinality constraints like 'finite' or 'countable'; it assumes a fixed first‑order language and standard Tarski semantics.

Semantic Tension

There is tension between 'axiomatizable' (axiomatizable by some first‑order set) and 'finitely axiomatizable' or 'axiomatizable by universal sentences'; different authors use 'elementary' to emphasize closure properties, which can obscure whether finitary/quantifier restrictions are meant.

Synthesis

An elementary class is the model‑theoretic embodiment of first‑order definability: a collection of structures precisely captured by a first‑order theory and stable under the elementary notions (equivalence and substructure) that preserve first‑order truth.