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well ordered set
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(Definition)
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A well-ordered set is a totally ordered set in which every nonempty subset has a least member.
An example of well-ordered set is the set of positive integers with the standard order relation
 , because any nonempty subset of it has least member. However,
 (the positive reals) is not a well-ordered set with the usual order, because $(0,1)=\{x:0<x<1\}$ is a nonempty subset but it doesn't contain a least number.
A well-ordering of a set $X$ is the result of defining a binary relation $\leq$ on $X$ to itself in such a way that $X$ becomes well-ordered with respect to $\leq$ .
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"well ordered set" is owned by drini. [ full author list (2) | owner history (2) ]
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Cross-references: binary relation, least number, contain, reals, relation, order, integers, positive, subset, totally ordered set
There are 44 references to this entry.
This is version 11 of well ordered set, born on 2001-10-17, modified 2005-07-26.
Object id is 271, canonical name is WellOrderedSet.
Accessed 21910 times total.
Classification:
| AMS MSC: | 06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order) | | | 03E25 (Mathematical logic and foundations :: Set theory :: Axiom of choice and related propositions) |
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Pending Errata and Addenda
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