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  "body": "\n## Definition\n\nModel theory is the part of mathematical logic whose objects are the following.\n\n1. Formal languages, most often the language of [[first-order logic|086QKPF0A6]] over a signature.\n2. [[Theories|92SWXST1ZM]], that is, sets of sentences of such a language.\n3. [[Structures|983GZBQYT4]] for the signature, and the relation of truth between structures and sentences, in particular the [[models|MVCH6N9SNW]] of theories.\n\nIt studies the classes of models of theories and the sets definable in structures.\n\n## Remarks\n\nExample: the statement that every field elementarily equivalent to $\\C$ is algebraically closed of characteristic $0$ is a result of model theory.\nIt concerns the class of models of the complete theory $\\mathrm{Th}(\\C)$.\n\nNon-example: the computation of the homology of a sphere is not a result of model theory, because it concerns no formal language and no relation of truth.\n\nClassical model theory works with first-order logic; other parts study [[infinitary logic|PQG7VAZ1R2]], finite structures or [[second-order logic|G6X7SZSNC5]].\n\n## Explanations\n\n### L1 human\n\nModel theory is the study of mathematical structures through the sentences that are true in them.\nIt sits between logic and algebra: its theorems are about formulas, and its applications are often about fields, groups and orders.\n",
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