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  "body": "\n## Summary\n\nA well-quasi-ordering is a [[preorder|0VABYWJWH9]] in which every infinite sequence $x_1, x_2, \\ldots$ has indices $i \u003c j$ with $x_i \\le x_j$.\nEquivalently, it is well-founded and has no infinite [[antichain|03HF498P1J]].\nWell-quasi-orders give finiteness results: every upward closed set is the upward closure of finitely many of its elements.\nBasic examples are $\\N^k$ with the product order ([[Dickson's lemma|NQD0Q9S9M7]]), finite words under subword embedding ([[Higman's lemma|GHK66XE1S8]]) and finite trees under embedding ([[Kruskal's tree theorem|XVHKCX4F5Y]]).\n\n## Explanations\n\n### L1 human\n\nIn a well-quasi-order, any endless list of elements must contain an element that is below a later element.\nYou cannot keep writing down new elements that avoid all earlier ones.\nFor pairs of natural numbers ordered coordinatewise, the list $(5, 0), (4, 1), (3, 2), (2, 3), (1, 4), (0, 5)$ has no pair below a later one, but no such list can go on forever.\nThe positive integers under divisibility fail this: the primes $2, 3, 5, 7, \\ldots$ never divide one another.\n",
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