Well-founded relation

Definition

Let $R$ be a binary relation on a set $A$. The relation $R$ is well-founded when each nonempty subset $S$ of $A$ has an element $m$ such that no $s \in S$ satisfies $s \mathrel{R} m$.

Remarks

The relation $<$ on $\mathbb{N}$ is well-founded. The relation $<$ on $\mathbb{Z}$ is not, because $\mathbb{Z}$ itself has no minimal element. If $R$ is well-founded, there is no sequence $x_0, x_1, x_2, \dots$ with $x_{n+1} \mathrel{R} x_n$ for each $n$. The converse needs the axiom of dependent choice. Well-founded relations allow proofs by transfinite induction.

Explanations

L1 human

In a well-founded relation, you cannot go down forever. Each strictly decreasing sequence of natural numbers stops after finitely many steps.