Definition:Strict Well-Ordering

Definition

Let $\prec$ be a strict total ordering on a class $A$.

Then $\prec$ is a strict well-ordering on $A$ iff $\prec$ is a foundational relation on $A$.

That is, expressed symbolically:

$\prec We A \iff \left({\prec Or A \land \prec Fr A}\right)$

Also see

• Well-Ordering

Note

Note that our definition does not require $A$ to be a set, as it can also be a proper class. This allows ordinals to be expressed in terms of strict well-orderings on $\in$, as the class of all ordinal numbers is a proper class.