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  "body": "\n## Statement\n\nSuppose that *S* and *T* are two first-order theories. As notation, let *S* ∪ *T* denote the smallest theory including both *S* and *T*; the signature of *S* ∪ *T* is the smallest one containing the signatures of *S* and *T*. Also let *S* ∩ *T* be the intersection of the languages of the two theories; the signature of *S* ∩ *T* is the intersection of the signatures of the two languages.\n\nLyndon's theorem says that if *S* ∪ *T* is unsatisfiable, then there is an interpolating sentence ρ in the language of *S* ∩ *T* that is true in all models of *S* and false in all models of *T*. Moreover, ρ has the stronger property that every relation symbol that has a positive occurrence in ρ has a positive occurrence in some formula of *S* and a negative occurrence in some formula of *T*, and every relation symbol with a negative occurrence in ρ has a negative occurrence in some formula of *S* and a positive occurrence in some formula of *T*.\n",
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    "name": "Lyndon's interpolation theorem",
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        "locator": "Craig interpolation, section Lyndon's interpolation theorem",
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  "name": "Lyndon's interpolation theorem",
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  "url": "https://math2.tamnd.com/wiki/lyndons-interpolation-theorem/",
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