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  "body": "\n## Definition\n\nLet $L$ be a signature and let $\\mathcal{M}$ and $\\mathcal{N}$ be $L$-[[structures|983GZBQYT4]].\nThe structures $\\mathcal{M}$ and $\\mathcal{N}$ are elementarily equivalent, written $\\mathcal{M} \\equiv \\mathcal{N}$, when for every $L$-[[sentence|DKFGQTRYZE]] $\\varphi$,\n$$\\mathcal{M} \\models \\varphi \\iff \\mathcal{N} \\models \\varphi.$$\n\n## Remarks\n\nEquivalently, $\\mathcal{M} \\equiv \\mathcal{N}$ when $\\mathrm{Th}(\\mathcal{M}) = \\mathrm{Th}(\\mathcal{N})$, where $\\mathrm{Th}$ is the set of sentences true in a structure.\n\nExample: $(\\Q, \u003c) \\equiv (\\R, \u003c)$.\nBoth are dense linear orders without endpoints, and the theory of dense linear orders without endpoints is complete.\n\nNon-example: $(\\Z, \u003c) \\not\\equiv (\\Q, \u003c)$.\nThe sentence $\\forall x \\, \\forall y \\, (x \u003c y \\to \\exists z \\, (x \u003c z \\land z \u003c y))$ is true in $(\\Q, \u003c)$ and false in $(\\Z, \u003c)$, with $x = 0$ and $y = 1$.\n\nIsomorphic structures are elementarily equivalent.\nThe example shows that the converse fails, since $\\Q$ and $\\R$ have different cardinalities.\n\n## Explanations\n\n### L1 human\n\nElementary equivalence says that first-order logic cannot tell two structures apart.\nIt is weaker than isomorphism: by the [[Löwenheim–Skolem theorem|484M0TZ1S7]], an infinite structure in a countable signature is elementarily equivalent to structures of every infinite size.\n",
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