Difference between revisions of "Well-Ordering Theorem"
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''Every set can be well-ordered''. | ''Every set can be well-ordered''. | ||
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| − | This result surprised | + | This result surprised [[mathematician]]s everywhere. Because the Well-Ordering Theorem is a direct consequence of the [[Axiom of Choice]], and no well-ordering relation has ever been explicitly constructed for [[uncountable set]]s, therefore many mathematicians have rejected the Axiom of Choice. |
[[Category: set theory]] | [[Category: set theory]] | ||
Revision as of 12:18, August 22, 2008
The Well-Ordering Theorem was proved by Zermelos in 1904, and it states:
Every set can be well-ordered.
This result surprised mathematicians everywhere. Because the Well-Ordering Theorem is a direct consequence of the Axiom of Choice, and no well-ordering relation has ever been explicitly constructed for uncountable sets, therefore many mathematicians have rejected the Axiom of Choice.