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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Thu, 7 Nov 2024 20:33:09 +0100
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On 07.11.2024 20:06, Jim Burns wrote:
 > On 11/7/2024 3:46 AM, WM wrote:

 >> That means by clever reordering them
 >> you can cover the whole positive axis
 >> except "boundaries".
 >
 > Yes.
 > In that clever re.ordering, not scrunched together,
 > the whole positive axis
 > is in the ε.cover or
 > in the boundary of the ε.cover.

It is impossible however to cover the real axis (even many times) by the 
intervals
  J(n) = [n - 1/10, n + 1/10].
No boundaries are involved because every interval of length 1/5 contains 
infinitely many rationals and therefore is essentially covered by 
infinitely many intervals of length 1/5 - if Cantor is right.

Regards, WM