Difference between revisions of "Well-Ordering Theorem"
(New page: The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states: <blockquote> Every set can be well-ordered. </blockquote> This result surprised mathematicians everywhere. B...) |
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The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states: | The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states: | ||
<blockquote> | <blockquote> | ||
| − | Every set can be well-ordered. | + | ''Every set can be well-ordered''. |
</blockquote> | </blockquote> | ||
| − | This result surprised | + | This result surprised [[mathematician]]s everywhere. The Well-Ordering Theorem is equivalent of the [[Axiom of Choice]], and no well-ordering relation has ever been explicitly constructed for [[uncountable set]]s. Thus, the mathematicians who reject the Axiom of Choice also reject this theorem. |
| − | [[Category: | + | |
| + | [[Category:Set Theory]] | ||
Latest revision as of 03:57, August 22, 2010
The Well-Ordering Theorem was proved by Zermelos in 1904, and it states:
Every set can be well-ordered.
This result surprised mathematicians everywhere. The Well-Ordering Theorem is equivalent of the Axiom of Choice, and no well-ordering relation has ever been explicitly constructed for uncountable sets. Thus, the mathematicians who reject the Axiom of Choice also reject this theorem.