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  "body": "\n## Summary\n\nFinite model theory is the part of [[model theory|HFTBG5PMXC]] that studies logics on finite [[structures|983GZBQYT4]].\nMany classical tools fail there; for example, the [[compactness theorem|1NR3HAWG9D]] does not hold when only finite structures are allowed.\nInstead the field uses combinatorial tools such as [[Ehrenfeucht–Fraïssé games|2RE7YAZK8R]] and locality arguments.\nIt is closely tied to computer science, through databases and descriptive complexity, where [[Fagin's theorem|BG8Q0BYSQ3]] matches a logic with a complexity class.\n\n## Explanations\n\n### L1 human\n\nA finite structure is, for example, a finite [[graph|00C0BR7D7H]] or a finite database table.\nFinite model theory asks which properties of such objects can be written as sentences of a given logic.\nA typical result: \"the graph is connected\" cannot be written as a single first-order sentence about finite graphs.\nThe proof uses a game between two players that compares two graphs move by move.\nQuestions of this kind relate the power of a logic to the computing time needed to check a property.\n",
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