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  "body": "\n## Summary\n\nThe Dedekind–MacNeille completion of a partially ordered set is the smallest [[complete lattice|D7A0ZGZT5S]] that contains it as an ordered subset.\nIt is built from cuts, pairs of sets of lower and upper bounds.\nThe real numbers are built in the same way from [[Dedekind cuts|0G3RRM41X9]] of the rationals.\nEvery existing supremum and infimum of the original poset is kept in the completion.\n\n## Explanations\n\n### L1 human\n\nThe rational numbers have gaps, such as the place where $\\sqrt{2}$ should be.\nDedekind filled these gaps by describing each missing number by the rationals below it.\nMacNeille did the same for any ordered set: each new element is a set of elements that is exactly the set of lower bounds of its own upper bounds.\nFor the rationals, the completion is the real line with $-\\infty$ and $+\\infty$ added.\nFor a finite lattice nothing new is added, because it is already complete.\n",
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