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| − | A place where nobody dared to go
| + | The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states: |
| − | The love that we came to know | + | <blockquote> |
| − | They call it Xanadu
| + | ''Every set can be well-ordered''. |
| − | | + | </blockquote> |
| − | And now
| + | 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. |
| − | Open your eyes and see
| + | [[Category: set theory]] |
| − | What we have made is real
| |
| − | We are in Xanadu
| |
| − | | |
| − | A million lights are dancing
| |
| − | And there you are
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| − | A shooting star
| |
| − | An everlasting world
| |
| − | And you're here with me
| |
| − | Eternally
| |
| − | | |
| − | Xanadu, Xanadu,
| |
| − | (now we are here)
| |
| − | In Xanadu
| |
| − | Xanadu, Xanadu,
| |
| − | (now we are here)
| |
| − | In Xanadu
| |
| − | | |
| − | Xanadu, your neon lights will shine
| |
| − | For you, Xanadu
| |
| − | | |
| − | The love
| |
| − | The echoes of long ago
| |
| − | You needed the world to know
| |
| − | They are in Xanadu
| |
| − | | |
| − | The dream
| |
| − | That came through a million years
| |
| − | That lived on through all the tears
| |
| − | It came to Xanadu
| |
| − | | |
| − | A million lights are dancing
| |
| − | And there you are
| |
| − | A shooting star
| |
| − | An everlasting world
| |
| − | And you're here with me
| |
| − | Eternally
| |
| − | | |
| − | Xanadu, Xanadu,
| |
| − | (now we are here)
| |
| − | In Xanadu
| |
| − | Xanadu, Xanadu,
| |
| − | (now we are here)
| |
| − | In Xanadu
| |
| − | | |
| − | Now that I'm here
| |
| − | Now that you're near in Xanadu
| |
| − | Now that I'm here
| |
| − | Now that you're near in Xanadu
| |
| − | Xanadu...
| |