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  "body": "\n## Definition\n\nLet $(A, \\le_A)$ and $(B, \\le_B)$ be [[totally ordered sets|0316AYH1SC]].\nThey have the same *order type* when there is an [[order isomorphism|0Y74KN99A5]] from $A$ onto $B$.\nThe *order type* of $(A, \\le_A)$, written $\\operatorname{type}(A)$, is the class of all totally ordered sets that have the same order type as $A$.\n\n## Remarks\n\nThe sets $\\N$ and $2\\N = \\set{0, 2, 4, \\dots}$ with their usual orders have the same order type, $\\omega$, by the isomorphism $n \\mapsto 2n$.\nThe sets $\\N$ and $\\Z$ do not: $\\N$ has a least element, $\\Z$ has none, and an order isomorphism maps a least element to a least element.\nHaving the same order type is an equivalence relation on totally ordered sets.\nEach well-ordered set is order isomorphic to a unique [[ordinal number|0RDJK7EQYY]], which is used to represent its order type.\nCommon notation: $\\omega$ for $\\N$, $\\omega^*$ for the reverse of $\\N$, $\\zeta$ for $\\Z$, $\\eta$ for $\\Q$ and $\\lambda$ for $\\R$.\nThe same definition makes sense for [[partially ordered sets|011C29Q1S1]].\n\n## Explanations\n\n### L1 human\n\nTwo ordered sets have the same order type when one can be relabelled into the other without changing the order.\nThe order type is the shape of the order.\n",
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