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Line 1: − + − <blockquote>+ − + − </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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The '''Well-Ordering Theorem''' was proved by Zermelos in 1904, and it states:
A place where nobody dared to go
The love that we came to know
''Every set can be well-ordered''.
They call it Xanadu
And now
Open your eyes and see
What we have made is real
We are in 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
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...