Difference between revisions of "Well-Ordering Theorem"

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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.
 
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.
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[[Category:Set Theory]]

Latest revision as of 03:57, 22 August 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.