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  "body": "\n## Statement\n\nLet $L$ be a signature and let $\\mathcal{M}$ be a [[substructure|676H7EGTKJ]] of an $L$-[[structure|983GZBQYT4]] $\\mathcal{N}$, with domains $M \\subseteq N$.\nThen $\\mathcal{M}$ is an [[elementary substructure|8WZF9SBP39]] of $\\mathcal{N}$ if and only if the following holds.\nFor every $L$-[[formula|HYJ10AF6RT]] $\\varphi(x_1, \\dots, x_n, y)$ and all $a_1, \\dots, a_n \\in M$, if $\\mathcal{N} \\models \\exists y \\, \\varphi(a_1, \\dots, a_n, y)$, then there is $b \\in M$ with $\\mathcal{N} \\models \\varphi(a_1, \\dots, a_n, b)$.\n\n## Remarks\n\nThe condition mentions truth only in $\\mathcal{N}$, which makes it easier to check than the definition.\nThe forward direction is direct: if $\\mathcal{N} \\models \\exists y \\, \\varphi(\\bar a, y)$, then $\\mathcal{M} \\models \\exists y \\, \\varphi(\\bar a, y)$, so some $b \\in M$ has $\\mathcal{M} \\models \\varphi(\\bar a, b)$, hence $\\mathcal{N} \\models \\varphi(\\bar a, b)$.\nFor the converse, one proves by induction on $\\varphi$ that $\\mathcal{M} \\models \\varphi(\\bar a)$ if and only if $\\mathcal{N} \\models \\varphi(\\bar a)$.\nAtomic formulas agree because $\\mathcal{M}$ is a substructure, the connectives are immediate, and the condition handles the existential quantifier.\n\nExample of use: in the proof of the downward [[Löwenheim–Skolem theorem|484M0TZ1S7]], one closes a subset of $N$ under witnesses for all existential formulas.\nThe test then shows that the result is an elementary substructure.\n\n## Explanations\n\n### L1 human\n\nThe test says: a substructure is elementary when every object that the big structure can describe with parameters from the small one has a witness already in the small one.\nThe positive integers inside $(\\N, \u003c)$ fail the test: the formula $y \u003c 1$ has the witness $0$ in $\\N$ but none among the positive integers.\n\n### L2 human\n\nThe Tarski–Vaught test is the standard tool to build elementary substructures.\nIt underlies the downward Löwenheim–Skolem theorem and the construction of Skolem hulls.\nCombined with automorphisms of $\\mathcal{N}$ that fix finitely many elements of $M$, it also proves results such as $(\\Q, \u003c) \\preceq (\\R, \u003c)$: if $\\mathcal{N} \\models \\exists y \\, \\varphi(\\bar a, y)$, an automorphism fixing $\\bar a$ moves a witness into $\\Q$.\n",
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