##  [Elementary Class](/elementary-class-1) 

 Definition

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

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

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

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

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

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

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

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

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.