Difference between revisions of "Well-Ordering Theorem"
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| − | + | The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states: | |
| − | The | + | <blockquote> |
| − | + | ''Every set can be well-ordered''. | |
| − | + | </blockquote> | |
| − | + | 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: set theory]] | |
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Revision as of 15:03, August 7, 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.