Definition:Strict Well-Ordering
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Definition
Definition 1
Let $\prec$ be a strict total ordering on a class $A$.
Then $\prec$ is a strict well-ordering on $A$ if and only if $\prec$ is a foundational relation on $A$.
That is, expressed symbolically:
- ${\prec} \mathrel{\operatorname{We}} A \iff \left({\prec \operatorname{Or} A \land {\prec} \mathrel{\operatorname{Fr}} A}\right)$
Definition 2
Let $\prec$ be a relation on $A$.
Then $\prec$ is a strict well-ordering of $A$ if and only if:
- $\prec$ connects $A$
- $\prec$ is well-founded. That is, whenever $b$ is a non-empty subset of $A$, $b$ has a $\prec$-minimal element.