4.5 Well-ordered sets

A relation $L \subseteq X \times X$ is well ordered on a set $X$ only if the following is true. The set $X$ is well ordered with respect to the relation $L$ :

• the relation is total
• $\forall Y \subseteq X: (Y \ne \{\}) \Rightarrow (\exists m \in Y: \forall v \in Y: ((m, v) \in L))$

For example, let us consider the relation less-than-or-equal-to ($\le$ ) and the set of all integers ( $\mathbb{I}$ ). The relation is total because $\forall i,j \in \mathbb{I}: (i \le j) \vee (j \le i)$ . On the other hand, $\forall Y \subseteq \mathbb{I}: (Y \ne \{\}) \Rightarrow (\exists m \in Y: \forall v \in Y: m \le v)$ .