Difference between revisions of "Well-ordering theorem"
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== See also == | == See also == | ||
| + | * [[Axiom of Choice]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
* [[Paradox]] | * [[Paradox]] | ||
* [[Ernst Zermelo]] | * [[Ernst Zermelo]] | ||
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== External links == | == External links == | ||
* [https://en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem] @ Wikipedia | * [https://en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem] @ Wikipedia | ||
Revision as of 19:09, 17 February 2016
In mathematics, the well-ordering theorem states that every set can be well-ordered.
Contents
Description
A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.
This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice.
Ernst Zermelo introduced the Axiom of Choice as an "unobjectionable logical principle" to prove the well-ordering theorem.
Transfinite induction
This is important because it makes every set susceptible to the powerful technique of transfinite induction.
Paradoxical consequences
The well-ordering theorem has consequences that may seem paradoxical, such as the Banach–Tarski paradox.
See also
External links
- Well-ordering theorem @ Wikipedia
