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  "body": "\n## Summary\n\nA complete ordered field is an [[ordered field|0P3VCBDEE2]] $F$ in which each nonempty subset that is bounded above has a [[supremum|0R3H5PANRQ]] in $F$.\nThe [[real numbers|08J2M1JYAG]] form a complete ordered field, by the [[completeness axiom|0AXWET80YW]].\nAny two complete ordered fields are isomorphic, by a [[unique isomorphism|0G8G1TH5WK]] that preserves the order.\nThus \"the complete ordered field\" is a characterization of $\\mathbb{R}$.\n\n## Remarks\n\nThe ordered field $\\mathbb{Q}$ is not complete.\nEach complete ordered field is [[Archimedean|0XH79S06J3]].\nOther forms of completeness of $\\mathbb{R}$ are equivalent for Archimedean ordered fields, for example the convergence of each Cauchy sequence.\n\n## Explanations\n\n### L1 human\n\nMany number systems allow addition, multiplication, division and comparison, for example $\\mathbb{Q}$ and $\\mathbb{R}$.\nAmong them, the real numbers are the only one that has no gaps.\nSo you can define the real numbers by this list of properties, without a construction.\n",
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