Re: Uncountable many reals without Cantor

From: Dave Seaman (dseaman_at_no.such.host)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 16:28:13 +0000 (UTC)

On Tue, 30 Nov 2004 10:10:14 -0600, David C Ullrich wrote:
> On Tue, 30 Nov 2004 14:56:30 +0000 (UTC), Dave Seaman
><dseaman@no.such.host> wrote:

>>On Tue, 30 Nov 2004 08:42:39 -0600, David C Ullrich wrote:
>>> On Tue, 30 Nov 2004 12:57:04 +0000 (UTC), Dave Seaman
>>><dseaman@no.such.host> wrote:
>>
>>>>On Tue, 30 Nov 2004 08:09:18 GMT, Jürgen R. wrote:
>>>>> David C. Ullrich <ullrich@math.okstate.edu> wrote:
>>>>
>>>>>>This argument _is_ much more complicated, if you include
>>>>>>the missing details. In particular you need a _proof_ of
>>>>>>the intuitively reasonable fact that if [0,1] is contained
>>>>>>in the union of countably many intervals I_n then
>>>>>>
>>>>>>(*) sum length(I_n) >= 1.
>>>>>>
>>>>>>How do you _prove_ that?
>>>>
>>>>> Assume the opposite, put the intervals end to end etc. This kind of
>>>>> thing is proven in the beginning of any Real Variables text, e.g.
>>>>> Royden. Where do you see a problem?
>>>>
>>>>Sounds like you want to use induction on the number of intervals.
>>>>Problem is, induction works only if the number of intervals is finite.
>>>>
>>>>Hint: that's where compactness comes in.
>>
>>> Amusing technicality:
>>
>>> In various places for various reasons we need to talk about
>>> coverings by half-open intervals instead of open intervals.
>>> Say [0,1) is the union of disjoint intervals [a_n, b_n).
>>> Then sum(b_n - a_n) = 1. You can't prove that (directly)
>>> by compactness, but you _can_ prove it by a simple
>>> transfinite induction!
>>
>>That's not quite the approach I had in mind. I'm sure you are aware of
>>this argument, but I wonder why we "need" to talk about half-open
>>intervals to show that m([0,1]) = 1. What I meant was to show the outer
>>measure of [0,1] is >= 1 by considering open covers, extracting finite
>>subcovers, and using ordinary induction over the number of intervals in
>>the subcover. Then you can show the outer measure is less than 1+epsilon
>>for any epsilon > 0.

> I understood all that - I wasn't disputing anything you said,
> just pointing out a _similar_ result where it's curious that
> one actually can use transfinite induction but not compactness
> (as opposed to the previous result where one can use compactness
> but not induction.)

Yeah, that's what I thought. I wouldn't have said anything if it hadn't
been for your choice of words in saying we "need" to talk about coverings
by half-open intervals. Didn't you recently chide someone else for
claiming that there is only one way to do something? (Or maybe two ways?) 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

Relevant Pages

  • Re: Uncountable many reals without Cantor
    ... >> Assume the opposite, put the intervals end to end etc. ... >> thing is proven in the beginning of any Real Variables text, ... induction works only if the number of intervals is finite. ... that's where compactness comes in. ...
    (sci.math)
  • Re: Uncountable many reals without Cantor
    ... >> Assume the opposite, put the intervals end to end etc. ... >> thing is proven in the beginning of any Real Variables text, ... induction works only if the number of intervals is finite. ... that's where compactness comes in. ...
    (sci.logic)
  • Re: Uncountable many reals without Cantor
    ... induction works only if the number of intervals is finite. ... Compactness for a topological space means the following: ... if for any collection of open sets U_i whose union is X, ...
    (sci.math)
  • Re: Uncountable many reals without Cantor
    ... induction works only if the number of intervals is finite. ... Compactness for a topological space means the following: ... if for any collection of open sets U_i whose union is X, ...
    (sci.logic)
  • Re: Uncountable many reals without Cantor
    ... >>subcovers, and using ordinary induction over the number of intervals in ... > one actually can use transfinite induction but not compactness ... > (as opposed to the previous result where one can use compactness ...
    (sci.logic)