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July 25, 2025, 09:42:50 pm
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Author Topic: Lethal Questions  (Read 1368 times)  Share 

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humph

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Lethal Questions
« on: May 17, 2008, 10:01:21 pm »
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Post your überhard maths questions here for all to compare!


I may be a bit bitter about that kinda thing right now. A question on my Analysis 2 assignment:

Prove there is an ordering of with the property that for each the set is at most countable.


... which you need to assume the continuum hypothesis to prove. Crazy stuff  ???
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kamil9876

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Re: Lethal Questions
« Reply #1 on: September 04, 2010, 11:34:41 pm »
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Doesn't the following ordering suffice?

iff is a non-negative rational number. Easy to check for transitivity and that other property whose name i keep forgetting. The countability follows from the countability of .

edit: just found out today it is called "anti-symmetry".
« Last Edit: September 06, 2010, 06:26:22 pm by kamil9876 »
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Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

Ahmad

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Re: Lethal Questions
« Reply #2 on: October 09, 2010, 10:09:35 am »
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I don't think that's a total order though

One idea is that there exists a minimal uncountable well-ordered set (by AC). Every section (the subsets in the initial post) is countable by minimality. We can find a bijection of this set with R assuming CH and the result follows.
« Last Edit: October 09, 2010, 10:12:43 am by Ahmad »
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kamil9876

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Re: Lethal Questions
« Reply #3 on: October 09, 2010, 09:38:37 pm »
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yeah it is not, he didn't say it has to be total though.
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Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."
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