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 $y\in X$, $x\ne y$. 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
$$(partialorder)\u27f9(totalorder)$$ |
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 |
---|---|
Canonical name | DefinitionOfWellOrderedSetAVariant |
Date of creation | 2013-03-22 18:04:47 |
Last modified on | 2013-03-22 18:04:47 |
Owner | yesitis (13730) |
Last modified by | yesitis (13730) |
Numerical id | 9 |
Author | yesitis (13730) |
Entry type | Derivation |
Classification | msc 03E25 |
Classification | msc 06A05 |
Related topic | SomethingRelatedToNaturalNumber |
Related topic | NaturalNumbersAreWellOrdered |