Definition 4.7.1.1. Let $(S, \leq )$ be a partially ordered set. We say that $(S,\leq )$ is well-founded if every nonempty subset $S_0 \subseteq S$ contains a minimal element: that is, an element $s \in S_0$ for which the set $\{ t \in S_0: t < s \} $ is empty.
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