Conflicting and Alternative Terminology
While the above is nowadays standard terminology in "infinite" model theory, the slightly different earlier definitions are still in use in finite model theory, where an elementary class may be called a Δ-elementary class, and the terms elementary class and first-order axiomatizable class are reserved for basic elementary classes (Ebbinghaus et al. 1994, Ebbinghaus and Flum 2005). Hodges calls elementary classes axiomatizable classes, and he refers to basic elementary classes as definable classes. He also uses the respective synonyms EC class and EC class (Hodges, 1993).
There are good reasons for this diverging terminology. 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. Finite model theory, on the other hand, deals almost exclusively with finite signatures. It is easy to see that for every finite signature σ and for every class K of σ-structures closed under isomorphism there is an elementary class of σ-structures such that K and contain precisely the same finite structures. Hence elementary classes are not very interesting for finite model theorists.
Read more about this topic: Elementary Class
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