Re: Peano's space-filling curve

From: Dave Seaman (dseaman_at_no.such.host)
Date: 06/12/04 

Date: Sat, 12 Jun 2004 14:52:44 +0000 (UTC)

On Sat, 12 Jun 2004 12:50:46 +0200, John Morgan wrote:

> Daniel Grubb <grubb@lola.math.niu.edu> wrote in message
> news:caa7t6$e5m$1@news.math.niu.edu...
>>
>>There are two standard approaches: Dedekind cuts, and
>> Cauchy sequences. Both are rather abstract. The
>> essential aspect of both is that 'holes' in the
>> collection of rational numbers have to be 'filled'.
>> They do so in different, but equivalent ways.

> Why_fill_them? I'm happy with rationals that have 'holes',
> because if these 'holes' cause me a problem, I switch to
> reals. If I understand D's cut, which I think I do, it
> assures us that sets of reals can divide into two without
> 'holes'.

That's like saying you are happy to do without computers, because if you
want to go on the Internet you can just use a web browser.

The whole point of "filling the holes" is to *construct* the real
numbers. Without Dedekind cuts (or Cauchy sequences, or some equivalent
construction) we don't have the real numbers at all. We might just
assume they exist, but we can't prove it without doing some work. 

-- 
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: Peanos space-filling curve
    ... > because if these 'holes' cause me a problem, ... > assures us that sets of reals can divide into two without ... Without Dedekind cuts (or Cauchy sequences, ...
    (sci.fractals)
  • CalvinBambi meets MoeBleeGodzilla (was: how to list all of the real numbers)
    ... What do you mean by "an expression of the reals"? ... Dedekind cuts in Z set theory provides for the definition of the ... Dedekind cuts or equivalence classes of Cauchy sequences? ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... What is the step of logic which leads one to seek an extention of the ...  But when they go from the rationals to the reals, ... class of Cauchy sequences that converge to that same point. ... built up, step by step, from rationals, to sequences ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... using the rationals as a starting point. ... that one could mix rational numbers with 'irrational Cauchy sequences', ... Do you mean each equivalence class is ... The top layer, so far, builds the reals from the rationals. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... mistaken you have what might be called second order Cauchy sequences. ... The next layer builds, say, the positive rationals from the naturals. ... The top layer, so far, builds the reals from the rationals. ... imperfection in the Cauchy sequence representation of a real number, ...
    (sci.math)