# definition of well ordered set, a variant

A well-ordered set is normally defined as a totally ordered set in which every nonempty subset has a least member, as the parent object does.

It is possible to define well-ordered sets as follows:

a well-ordered set $X$ is a partially ordered set in which every nonempty subset of $X$ has a least member.

To justify the alternative, we prove that every partially ordered set $X$ in which every nonempty subset has a least member is total:

let $x\in X$ and $y\in X$, $x\neq y$. Now, $\{x,y\}$ has a least member, a fortiori, $x,y$ are comparable. Hence, $X$ is totally ordered.

The alternative has the benefit of being a stronger statement in the sense that

 $(partial\;order)\Longrightarrow(total\;order)$

given that every nonempty subset has a least member.

## References

• 1 Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.
• 2 Jech, T., Set Theory, 3rd millennium ed., Springer, 2002.
Title definition of well ordered set, a variant DefinitionOfWellOrderedSetAVariant 2013-03-22 18:04:47 2013-03-22 18:04:47 yesitis (13730) yesitis (13730) 9 yesitis (13730) Derivation msc 03E25 msc 06A05 SomethingRelatedToNaturalNumber NaturalNumbersAreWellOrdered