Difference between revisions of "Well-Ordering Theorem"
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(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 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 set]]s, therefore many mathematicians have rejected the | + | 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 set]]s, therefore many mathematicians have rejected the Axiom of Choice. |
[[Category: set theory]] | [[Category: set theory]] | ||
Revision as of 10:33, April 2, 2007
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.
