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  "body": "\n## Definition\n\nLet $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.\nThe relation $R$ is well-founded when each nonempty subset $S$ of $A$ has an element $m$ such that no $s \\in S$ satisfies $s \\mathrel{R} m$.\n\n## Remarks\n\nThe relation $\u003c$ on $\\mathbb{N}$ is well-founded. The relation $\u003c$ on $\\mathbb{Z}$ is not, because $\\mathbb{Z}$ itself has no minimal element.\nIf $R$ is well-founded, there is no sequence $x_0, x_1, x_2, \\dots$ with $x_{n+1} \\mathrel{R} x_n$ for each $n$. The converse needs the axiom of dependent choice.\nWell-founded relations allow proofs by [[transfinite induction|0ZHQMQ2C1N]].\n\n## Explanations\n\n### L1 human\n\nIn a well-founded relation, you cannot go down forever.\nEach strictly decreasing sequence of natural numbers stops after finitely many steps.\n",
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