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  "body": "\n## Summary\n\nAn elementary class is the class of all [[models|MVCH6N9SNW]] of a first-order [[theory|92SWXST1ZM]] in a fixed signature.\nGroups, rings, fields of characteristic $0$ and dense linear orders are elementary classes.\nAn elementary class is closed under [[elementary equivalence|P3B1V182N4]] and under ultraproducts.\nBy the [[compactness theorem|1NR3HAWG9D]], many natural classes, such as torsion groups, are not elementary.\nUniversal classes, universal Horn classes, quasivarieties and varieties are elementary classes with axioms of special shapes.\n\n## Explanations\n\n### L1 human\n\nPick a list of first-order axioms, such as the group axioms.\nThe elementary class of the list is everything that satisfies all of them.\nFirst-order axioms may quantify over elements but not over sets of elements or over natural numbers.\nSo \"every element has finite order\" cannot be written as one first-order axiom, and torsion groups are not an elementary class.\n",
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