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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a partially ordered set.\nThe set $P$ satisfies the ascending chain condition if every sequence $a_1 \\le a_2 \\le a_3 \\le \\dots$ of elements of $P$ is eventually constant.\nThat is, there is an integer $N$ with $a_n = a_N$ for all $n \\ge N$.\n\n## Remarks\n\nEquivalently, every nonempty subset of $P$ has a maximal element. This equivalence uses the axiom of dependent choice.\nThe set $\\mathbb{N}$ with its usual order does not satisfy the condition. With the reverse order, it does.\nFor the set of the ideals of a ring, ordered by inclusion, the condition defines [[Noetherian rings|0MVH4Y1CRZ]].\nThe ideals of $\\mathbb{Z}$ satisfy the condition: a chain $(a_1) \\subseteq (a_2) \\subseteq \\dots$ means $a_2 \\mid a_1$, $a_3 \\mid a_2$, and so on.\n\n## Explanations\n\n### L1 human\n\nThe condition says that you cannot climb forever.\nEvery increasing chain stops growing after a finite number of steps.\nFor the ideals of the integers, a strictly increasing chain like $(12) \\subset (6) \\subset (3) \\subset (1)$ must end.\n",
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