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  "body": "\n## Summary\n\nLet $G$ be an abelian [[linearly ordered group|P2E8BNYC7B]].\nTwo nonzero elements of $G$ are Archimedean equivalent when each is bounded by a multiple of the other, in absolute value.\nThe equivalence classes measure the size of elements up to finite factors, in the way orders of magnitude do.\nThey form a totally ordered set, and the group is Archimedean exactly when it has at most one class of nonzero elements.\nThese classes index the coordinates in the [[Hahn embedding theorem|FVPXXFNVQY]].\n\n## Explanations\n\n### L1 human\n\nCompare the sizes of things up to any finite factor.\nA millimetre and a kilometre are in the same class, since a million millimetres make a kilometre.\nIn a group with infinitely large elements, some element is larger than every multiple of another one, and then they lie in different classes.\nIn $\\Z^2$ with the dictionary order, $(0, 1)$ and $(0, 5)$ are in the same class, but $(1, 0)$ is infinitely larger than both.\n",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RQ4/RQ4X0EH6NH.md",
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  "url": "https://tamnd.github.io/math2/wiki/archimedean-equivalence-classes/",
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