Path: ...!weretis.net!feeder9.news.weretis.net!i2pn.org!i2pn2.org!.POSTED!not-for-mail
From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Fri, 8 Nov 2024 07:28:53 -0500
Organization: i2pn2 (i2pn.org)
Message-ID: <d780ead415ff3a62ccd9b606bcd743fea3d8002c@i2pn2.org>
References: <vg7cp8$9jka$1@dont-email.me>
 <0e67005f-120e-4b3b-a4d2-ec4bbc1c5662@att.net> <vgab11$st52$3@dont-email.me>
 <ecffc7c0-05a2-42df-bf4c-8ae3c2f809d6@att.net> <vgb0ep$11df5$4@dont-email.me>
 <35794ceb-825a-45df-a55b-0a879cfe80ae@att.net> <vgfgpo$22pcv$1@dont-email.me>
 <40ac3ed2-5648-48c0-ac8f-61bdfd1c1e20@att.net> <vgg57o$25ovs$2@dont-email.me>
 <71fea361-0069-4a98-89a4-6de2eef62c5e@att.net> <vggh9v$27rg8$3@dont-email.me>
 <ff2c4d7c-33b4-4aad-a6b2-88799097b86b@att.net> <vghuoc$2j3sg$1@dont-email.me>
 <d79e791d-d670-4a5a-bd26-fdf72bcde6bc@att.net> <vgj4lk$2ova9$3@dont-email.me>
 <f154138e-4482-4267-9332-151e2fd9f1ba@att.net> <vgkoi7$b5pp$1@solani.org>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Fri, 8 Nov 2024 12:28:54 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="1517201"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
Content-Language: en-US
In-Reply-To: <vgkoi7$b5pp$1@solani.org>
X-Spam-Checker-Version: SpamAssassin 4.0.0
Bytes: 3406
Lines: 41

On 11/8/24 5:18 AM, WM wrote:
> On 08.11.2024 00:29, Jim Burns wrote:
>  > On 11/7/2024 2:33 PM, WM wrote:
> 
>  >> It is impossible however to cover
>  >> the real axis (even many times) by
>  >> the intervals
>  >> J(n) = [n - 1/10, n + 1/10].
>  >
>  > Those are not the cleverly.re.ordered intervals.
> They are the intervals that we start with.
>  >> 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.
>  >
>  > I haven't claimed anything at all about
>  > your all.1/5.length intervals.
> Then consider the two only alternatives: Either by reordering (one after 
> the other or simultaneously) the measure of these intervals can grow 
> from 1/10 of the real axis to infinitely many times the real axis, or not.
> 
> My understanding of mathematics and geometry is that reordering cannot 
> increase the measure (only reduce it by overlapping). This is a basic 
> axiom which will certainly be agreed to by everybody not conditioned by 
> matheology. But there is also an analytical proof: Every reordering of 
> any finite set of intervals does not increase their measure. The limit 
> of a constant sequence is this constant however.
> 
> This geometrical consequence of Cantor's theory has, to my knowledge, 
> never been discussed. By the way I got the idea after a posting of 
> yours: Each of {...,-3,-2,-1,0,1,2,3,...} is the midpoint of an interval.
> 
> Regards, WM
> 

which makes the error that the properties of finite objects apply to the 
infinite objects, which isn't true, and what just breaks your logic.

You take it as a given, but that just means that your logic is unable to 
actually handle the infinite.