# Definition:Strict Strong Well-Ordering

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## Contents

## 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.

## Sources

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