Re: Uncountable many reals without Cantor

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

Date: Tue, 30 Nov 2004 17:26:56 +0000 (UTC)

On Tue, 30 Nov 2004 16:28:13 +0000 (UTC), Dave Seaman wrote:
> On Tue, 30 Nov 2004 10:10:14 -0600, David C Ullrich wrote:

>> 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.)

One other thought -- you may not be using compactness in that part of the
argument, but you do need some special property of the reals in order to
conclude that you only need to consider end-to-end placement of the
intervals in the first place. Consider covering the rationals in [0,1]
by collections of half-open intervals, for example. 

-- 
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
    ... >> one actually can use transfinite induction but not compactness ... >> (as opposed to the previous result where one can use compactness ... by collections of half-open intervals, ... Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ...
    (sci.logic)
  • Re: Uncountable many reals without Cantor
    ... >That's not quite the approach I had in mind. ... >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)
  • Re: Uncountable many reals without Cantor
    ... >That's not quite the approach I had in mind. ... >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.math)