axiomatizable class
Let $L$ be a first order language and $T$ a theory in $L$. Recall that a model $M$ is an $L$structure^{} such that $M$ satisfies every sentence^{} in $T$. We say that the structure $M$ is a model of $T$. Let us write $\mathrm{Mod}(T)$ the class of all $L$structures that are models of $T$.
Definition. A class $K$ of $L$structures is said to be axiomatizable if there is a theory $T$ such that $K=\mathrm{Mod}(T)$. Furthermore, $K$ is a finitely axiomatizable^{} or elemenary class if $T$ is finite.
For example, the class of groups is elementary (and hence axiomatizable), because the set of group axioms is finite. However, the class of infinite groups is axiomatizable but not elementary. Similarly, the class of $R$modules is elementary iff $R$ is finite. The class of locally finite groups is an example of a nonaxiomatizable class.
Remarks.

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$K$ is an elementary class iff there is a sentence $\phi $ such that $K=\mathrm{Mod}(\{\phi \})$, for sentences ${\phi}_{1},\mathrm{\dots},{\phi}_{n}$ can be combined to form ${\phi}_{1}\wedge \mathrm{\cdots}\wedge {\phi}_{n}$, which is also a sentence since it has no free variables^{}.

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A class is axiomatizable iff it is an intersection^{} of elementary classes. As such elementary class is sometimes abbreviated EC, and axiomatizable class EC${}_{\mathrm{\Delta}}$, where $\mathrm{\Delta}$ means is another symbol for intersection.

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A caution to the reader: some authors call an elementary class an axiomatizable class that is defined here.
Title  axiomatizable class 

Canonical name  AxiomatizableClass 
Date of creation  20130322 17:34:38 
Last modified on  20130322 17:34:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03C52 
Synonym  axiomatisable class 
Synonym  finitely axiomatizable 
Synonym  finitely axiomatisable 
Synonym  EC 
Synonym  EC${}_{\mathrm{\Delta}}$ 
Related topic  Supercategories3 
Related topic  AxiomaticAndCategoricalFoundationsOfMathematicsII2 
Defines  elementary class 