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  "body": "\n## Example\n\nA [[linear order|0FRXBVVR03]] $(D, \u003c)$ is *dense* if for all $a \u003c b$ in $D$ there is $c \\in D$ with $a \u003c c \u003c b$.\nThe theory $\\mathrm{DLO}$ of dense linear orders without endpoints is a theory in the signature $\\set{\u003c}$.\nIt says that $\u003c$ is a dense linear order on a set with at least two elements.\nIt also says that there is no least and no greatest element.\nThe [[rational numbers|0NCJ5WXN6Z]] and the real numbers with their usual order are models.\n\n$\\mathrm{DLO}$ is $\\aleph_0$-categorical: any two countable models are [[order isomorphic|0D3WA36K2Q]].\nCheck, by Cantor's back-and-forth argument: list the elements of two countable models $A$ and $B$, and build a chain of finite order-preserving partial maps.\nAt odd steps, send the next element of $A$ to an element of $B$ in the same position relative to the finite image.\nThis element exists because $B$ is dense and has no endpoints.\nAt even steps, do the same from $B$ back to $A$.\nThe union of the chain is an isomorphism from $A$ onto $B$.\n\n## Remarks\n\nSince $\\mathrm{DLO}$ has no finite models, the [[Łoś–Vaught test|ZR7Y5C235P]] shows that it is complete; it is also decidable.\nIt has [[quantifier elimination|7FDDXV3JK7]].\nIt is not $\\kappa$-categorical for any uncountable $\\kappa$.\nFor $\\kappa = 2^{\\aleph_0}$, compare $\\R$ with the order $\\R + \\Q$, a copy of $\\R$ followed by a copy of $\\Q$.\nBoth are dense without endpoints and have size $2^{\\aleph_0}$.\nThey are not isomorphic.\nIn $\\R$ every nonempty set with an upper bound has a least upper bound.\nIn $\\R + \\Q$ the copy of $\\R$ is bounded above but has no least upper bound.\n\n## Explanations\n\n### L1 human\n\nIn a dense order, between any two elements there is a third, like the rational numbers on the number line.\nEvery countable dense order without a first or last element is a copy of the rational numbers.\nThis makes the theory of such orders one of the simplest examples in model theory.\nIt is complete, it has only one countable model, and every formula reduces to comparisons between the variables.\n",
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