Definition:Well-Founded Ordered Set
(Redirected from Definition:Well-Founded)
Jump to navigation
Jump to search
- 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.
The term well-founded can equivalently be said to apply to the ordering $\preceq$ itself rather than to the ordered set $\struct {S, \preceq}$ as a whole.
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