Definition:Strict Strong Well-Ordering

Definition

Let $A$ be a class.

Let $\mathcal R$ be a relation on $A$.

Then $\mathcal R$ is a strict strong well-ordering of $A$ if and only if:

$\mathcal R$ connects $A$
$\mathcal R$ is strongly well-founded. That is, whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.

Also known as

1955: John L. Kelley: General Topology calls this a well-ordering, but we use that term in a slightly different sense.

Remarks

The term "strict strong well-ordering" was invented for $\mathsf{Pr} \infty \mathsf{fWiki}$ to distinguish between this notion and the weaker notion of a strict well-ordering.

In the presence of the Axiom of Foundation, they are equivalent.