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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|071DDFVG4J]] with a total order that makes it a [[linearly ordered group|1P8087YGYQ]].\nLet $\\Omega$ be the set of [[Archimedean classes|C8VE2SEVEB]] of the nonzero elements of $G$.\nOrder $\\Omega$ so that $[a] \\prec [b]$ when $a$ is infinitely larger than $b$.\nLet $H$ be the set of maps $f \\colon \\Omega \\to \\R$ whose support $\\set{\\omega : f(\\omega) \\ne 0}$ is [[well-ordered|0F1D6PK9RZ]] by $\\prec$.\nMake $H$ a group by pointwise addition, and set $f \u003e 0$ when $f(\\omega_0) \u003e 0$ for the least element $\\omega_0$ of the support of $f$.\nThen $H$ is a linearly ordered abelian group, and there is an injective [[group homomorphism|0HV9Y7WG5R]] $\\varphi \\colon G \\to H$ with $a \\le b$ if and only if $\\varphi(a) \\le \\varphi(b)$.\n\n## Remarks\n\nHahn proved the theorem in 1907.\nFor a nonzero $a \\in G$, let $C_a$ be the smallest convex subgroup containing $a$ and $D_a$ the largest convex subgroup not containing $a$.\nThe quotient $C_a / D_a$ depends only on the class of $a$.\nThis quotient is Archimedean, so [[Hölder's theorem|CDM40MT3HZ]] embeds it in [[the reals|08J2M1JYAG]].\nThe proofs then glue these embeddings together, using divisible hulls and Zorn's lemma.\nExample: $\\Z^2$ with the lexicographic order has two nonzero classes, and its image in $H \\cong \\R^2$ is the obvious one with the lexicographic order.\nWhen $G$ is Archimedean, $\\Omega$ has one element and the theorem reduces to Hölder's theorem.\n\n## Explanations\n\n### L1 human\n\nAny way of ordering a commutative group so that addition respects the order can be described with real numbers arranged by size of magnitude.\nEach element becomes a list of real coordinates, one for each order of magnitude, read from the largest magnitude down.\nTwo elements are compared by the first coordinate where they differ.\n\n### L2 human\n\nThe theorem says that the groups $H$, called Hahn groups, are universal for linearly ordered abelian groups with a given set of Archimedean classes.\nIt is the group counterpart of the fact that the [[Hahn series|YQP8V1KKMW]] fields are large enough to contain many valued fields.\n",
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