# Definition:Strict Strong Well-Ordering

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## Definition

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.

## Sources

- 1955: John L. Kelley:
*General Topology*: Appendix: Definition $87$