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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]], and write $x \u003c y$ when $x \\le y$ and $x \\ne y$.\nThe order is *well-founded* when every nonempty subset $A \\subseteq P$ has a minimal element, that is, an element $m \\in A$ such that no $a \\in A$ satisfies $a \u003c m$.\n\n## Remarks\n\nExample: the set $\\N^2$ with the product order, $(a, b) \\le (c, d)$ when $a \\le c$ and $b \\le d$, is well-founded.\nCheck: in a nonempty $A \\subseteq \\N^2$, take the least first coordinate $a_0$ occurring in $A$, and then the least $b_0$ with $(a_0, b_0) \\in A$; no element of $A$ is strictly below $(a_0, b_0)$.\n\nNon-example: the set $\\set{1/n : n \\ge 1}$ of rational numbers with the usual order is not well-founded, since the whole set has no minimal element.\n\nThe order is well-founded exactly when the strict order $\u003c$ is a [[well-founded relation|0DQPNJNPYK]].\nWith the axiom of dependent choice, the condition is equivalent to the absence of an infinite sequence $x_0 \u003e x_1 \u003e x_2 \u003e \\cdots$.\nThe same definition applies to a preorder, with $x \u003c y$ meaning $x \\le y$ and not $y \\le x$.\n",
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