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  "body": "\n## Definition\n\nA Dedekind cut is a pair $(A, B)$ of subsets of the [[rational numbers|0NCJ5WXN6Z]] $\\mathbb{Q}$ with these properties:\n\n1. $A$ and $B$ are nonempty, $A \\cap B = \\emptyset$ and $A \\cup B = \\mathbb{Q}$;\n2. if $a \\in A$, $q \\in \\mathbb{Q}$ and $q \u003c a$, then $q \\in A$;\n3. $A$ has no greatest element.\n\n## Remarks\n\nThe set $A$ determines the cut, because $B = \\mathbb{Q} \\setminus A$. Thus some authors call $A$ itself the cut.\nThe cut with $A = \\{q \\in \\mathbb{Q} : q \u003c 0 \\text{ or } q^2 \u003c 2\\}$ corresponds to $\\sqrt{2}$. Here $B$ has no least element.\nThe set of all cuts, ordered by inclusion of the lower sets, is a [[complete ordered field|0A9E6X0S3E]] with suitable operations.\n\n## Explanations\n\n### L1 human\n\nCut the rational number line at one point. Put the numbers to the left in $A$ and the numbers to the right in $B$.\nWhen the cut is at a gap of $\\mathbb{Q}$, such as $\\sqrt{2}$, the cut itself stands for the missing number.\n",
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