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Elementary class
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== Conflicting and alternative terminology == While the above is nowadays standard terminology in [[model theory|"infinite" model theory]], the slightly different earlier definitions are still in use in [[finite model theory]], where an elementary class may be called a '''Ξ-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<math>_\Delta</math> class''' and '''EC class''' (Hodges, 1993). There are good reasons for this diverging terminology. The [[signature (logic)|signature]]s that are considered in general model theory are often infinite, while a single [[first-order logic|first-order]] [[sentence (mathematical logic)|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 Ο and for every class ''K'' of Ο-structures closed under isomorphism there is an elementary class <math>K'</math> of Ο-structures such that ''K'' and <math>K'</math> contain precisely the same finite structures. Hence, elementary classes are not very interesting for finite model theorists.
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