Timeline for Proofs of the uncountability of the reals
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Dec 22, 2010 at 16:38 | comment | added | Unknown | I am sorry I have the book in hard, but can't find where the proof is. Could you tell me where that is? | |
| Dec 5, 2010 at 5:36 | comment | added | roy smith | this proof appears in hardy's pure mathematics, about 100 years ago. | |
| Nov 27, 2010 at 18:06 | comment | added | gowers | Maybe you've convinced me. One can take the family you describe and define x to be the supremum over all reals x such that [0,x] can be covered by finitely many of its members. Such an x can't be covered itself (since the sets are open). This is just the proof of Heine-Borel in this special case. Also, your lemma gives us that x can't be 1. So one ends up using the least upper bound axiom instead of the nested-intervals property, which is perhaps enough to qualify the argument as genuinely different. | |
| Nov 26, 2010 at 0:38 | comment | added | Pablo Shmerkin | I edited my post with a sketch of a proof that $[0,1]$ has outer measure $1$, modulo compactness of closed intervals. I do not see that I'm using a diagonal/nested intervals argument, but I could be totally wrong. The usual proof of Heine-Borel also doesn't appear to me to use a diagonal argument (see e.g. math.utah.edu/~bobby/3210/heine-borel.pdf). At no point in the proof there is an enumeration of an infinite set. | |
| Nov 26, 2010 at 0:30 | history | edited | Pablo Shmerkin | CC BY-SA 2.5 |
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| Nov 25, 2010 at 22:15 | comment | added | gowers | I'd be quite surprised if you could prove that that family you define doesn't cover the whole of [0,1] without using something very like the nested-intervals version of the diagonal argument. (That is, I'd like to know more about your proof that [0,1] has outer measure 1.) | |
| Nov 23, 2010 at 1:05 | history | edited | Pablo Shmerkin | CC BY-SA 2.5 |
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| Nov 23, 2010 at 0:58 | history | edited | Pablo Shmerkin | CC BY-SA 2.5 |
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| Nov 23, 2010 at 0:50 | history | answered | Pablo Shmerkin | CC BY-SA 2.5 |