Definition:Well-Founded Ordered Set
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- Not to be confused with Definition:Well-Founded Set.
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Then $\struct {S, \preceq}$ is well-founded if and only if it satisfies the minimal condition:
- Every non-empty subset of $S$ has a minimal element.
That is, if the ordering $\preceq$ is a well-founded relation.
Also see
Stronger properties
Generalization
Sources
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations