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:"Cantor's first proof" is new to me, and I have to say it's delightful. I agree that mathematicians generally believe the diagonal argument to be Cantor's first. However, I'm not completely convinced that this isn't really a diagonal argument in disguise. I need to think about this a bit. [[User:Dmharvey|Dmharvey]] [[Image:User_dmharvey_sig.png]] [[User talk:Dmharvey|Talk]] 22:45, 6 Jun 2005 (UTC)
== Simplifications ==
I'm hesitant to make changes to the statement of the theorem and its proof, since we may want to retain historical accuracy, and I don't have access to Cantor's formulation. There's five things I would change:
# '''R''' needs to be non-empty which is currently not required. (but see 2 below)
# if we require that '''R''' have at least ''two'' points, then we can get rid of the endpointlessness requirement. (The first step of the proof would then read "pick the first element of the sequence that's not the largest element of '''R'''"; everything else stays the same.)
# The proof emphasizes "greater than the one considered in the previous step" twice, but that's not needed; just always pick the first member of the sequence that works.
# The proof should make a bit more explicit how the existence of ''c'' follows from the given properties of '''R'''. I.e.: How the sets ''A'' and ''B'' are defined.
# The proof should make explicit how the density property is used.
What do people think about these changes? [[User:AxelBoldt|AxelBoldt]] 17:49, 30 March 2006 (UTC)
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