Definition:Strict Strong Well-Ordering

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Let $A$ be a class.

Let $\RR$ be a relation on $A$.

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

$\RR$ connects $A$
$\RR$ is strongly well-founded. That is, whenever $B$ is a non-empty subclass of $A$, $B$ has a strictly minimal element under $\RR$ .

Also known as

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

Linguistic Note

The term Strict Strong Well-Ordering was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ to distinguish between this notion and the weaker notion of a strict well-ordering.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

In the presence of the Axiom of Foundation, a strict strong well-ordering and a strict well-ordering are equivalent.