well-founded relation

A binary relation $R$ on a class (http://planetmath.org/Class) $X$ is well-founded if and only if

• each nonempty subclass of $X$ contains an $R$-minimal element and,

• for each $x\in X$, $\{y\mid yRx\}$ is a set.

The notion of a well-founded relation is a generalization of that of a well-ordering relation: proof by induction and definition by recursion may be carried out over well-founded relations.