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  "body": "\n## Statement\n\nLet $(A, \u003c)$ and $(B, \u003c)$ be [[strict total orders|C9GDH7264N]] with these properties.\n\n1. $A$ and $B$ are nonempty and [[countable|0ZG61TFBES]].\n2. Both orders are [[dense|JB9HJ9CPEN]].\n3. Neither $A$ nor $B$ has a least element or a greatest element.\n\nThen there is an [[order isomorphism|0Y74KN99A5]] from $A$ onto $B$.\nIn particular, each such order is order isomorphic to $(\\Q, \u003c)$.\n\n## Remarks\n\nA standard proof is the back-and-forth method.\nList $A = \\set{a_0, a_1, \\dots}$ and $B = \\set{b_0, b_1, \\dots}$, and build finite order-preserving partial maps step by step.\nAt even steps, the first element of $A$ not yet in the domain gets an image.\nAt odd steps, the first element of $B$ not yet in the range gets a preimage.\nDensity and the lack of endpoints guarantee that a suitable partner exists at each step.\nThe union of the finite maps is the isomorphism.\nFor example, the dyadic rationals $\\set{m / 2^k : m \\in \\Z, k \\in \\N}$ and the rationals in the open interval $(0, 1)$ are both order isomorphic to $\\Q$.\nCountability is needed: $\\R$ satisfies the other hypotheses but is not order isomorphic to $\\Q$, because $\\R$ is uncountable.\nThe condition on endpoints is needed: $\\Q \\cap [0, 1]$ has a least element and $\\Q$ has none.\nDensity is needed: $\\Z$ is countable without endpoints but not dense.\nIn model theory, the theorem says that the theory of dense linear orders without endpoints is $\\aleph_0$-categorical.\n\n## Explanations\n\n### L1 human\n\nThe rationals have a very particular shape: countably many points, no ends, and always something between two points.\nThe theorem says this shape is unique.\nAny other countable line with no ends and no gaps of the \"next element\" kind can be matched point by point with the rationals, keeping the order.\nSo the rationals between $0$ and $1$ and all the rationals have the same order, even though one sits inside the other.\n\n### L2 human\n\nThe theorem is the basic example of the back-and-forth method, which is used widely in model theory.\nIt shows that the [[order type|28PE4GRFD7]] $\\eta$ of $\\Q$ is the unique countable dense order type without endpoints.\nEvery countable total order embeds into $\\Q$; the proof is the \"forth\" half of the same argument.\nThe analogous question for uncountable orders is much harder; [[Suslin's problem|D1T39KHPDN]] is a famous case.\n",
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