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 Posted: Fri Jun 08, 2007 12:47 pm
So, for some reason, I'm irked by proofs by contradiction. While I'm willing to learn them when they seriously reduce the complexity of a proof, I always make it a point to try to write a different proof for the same [theorem, lemma, deelie] using something else.
Right now, I'm stuck on providing a different proof for this (very basic) topological theorem:
"If E is an infinite subset of a compact metric space K, then E has a limit point in K."
I realize that you can replace "compact metric space" by "compact space" and the theorem is still true, but considering I can't even provide a different proof for the metric space case...grr. gonk
Any ideas? I realize this is a pretty basic theorem, so I won't be surprised when [random person] provides really elegant, simple proof.
[Edit] For reference, the proof by contradiction:
"S'pose that no point of K is a limit point of E. Then for every point q in K, there is some neighborhood V_q such that the intersection of V_q with E is either empty (if q is not in E) or q (if q is in E). The collection of neighborhoods V_q is an open cover for K, but since E is infinite, there is no finite subcollection which covers E (and hence K). Therefore: if a set K contains an infinite subset E which has no limit points in K, K is not compact; thus if K is compact, every infinite subset of K has a limit point in K.
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Posted: Fri Jun 08, 2007 2:08 pm
To that effect, I also had a problem proving:
"Every nonempty, perfect subset of the real numbers is uncountable."
The proof did...strange things. It constructed a collection of sets based on a countable number of points of a nonempty perfect set, which violates the theorem that
"If {K} is a collection of compact sets such that the intersection of every finite subcollection of {K} is nonempty, then the intersection of all of {K} is nonempty."
I wasn't quite sure what this proved. Does this show that no countable subset of the real line has limit points? That every countable subset of the real line has isolated points? Confused...
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Posted: Tue Jun 12, 2007 1:13 am
I've had a bit of a think and I can't come up with an argument either. However, there is a whole school of mathematics that buys the same idea that you do, and you might be able to find something if you look around. Here's an article about mathematical contructivism. http://en.wikipedia.org/wiki/Mathematical_constructivism
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Posted: Thu Jun 14, 2007 7:15 am
I hadn't really thought of constructivism as an entire school of philosophy. Thanks for the article. For me, it's really more of a personal thing. I never feel like proofs by contradiction illuminate anything; after reading a proof by contradiction, I don't feel like I've learned anything more. And that is extremely frustrating.
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Posted: Wed Jun 27, 2007 9:26 pm
Well by proof of contradiction you want to prove that there is no case where:
"E has a limit point in K and E is not an infinite subset of a compact metric space K"
or by contrapositive, you want to prove that:
"If E does not have a limit point in K then E is not an infinite subset of a compact metric space K"
Hope that helps.
((Personally I think proof by contradiction is fun))
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