# Definition:Well-Founded Ordered Set

(Redirected from Definition:Well-Founded)

*Not to be confused with Definition:Well-Founded Set.*

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ 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 as a whole.

## Remark

The term well-founded is also commonly used for foundational relations, which are closely related to, but different from, well-founded orderings.

## Also see

### Stronger properties

### Generalization

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Relations