Path: ...!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Thu, 7 Nov 2024 10:28:17 +0100
Organization: A noiseless patient Spider
Lines: 33
Message-ID: <vgi17g$2j3sg$5@dont-email.me>
References: <vg7cp8$9jka$1@dont-email.me>
 <0e67005f-120e-4b3b-a4d2-ec4bbc1c5662@att.net> <vga5mb$st52$1@dont-email.me>
 <vga7qi$talf$1@dont-email.me> <03b90d6c-fff1-411d-9dec-1c5cc7058480@tha.de>
 <380a63a4ba5e4206504c27b06c9f2a4b6bf3c02d@i2pn2.org>
 <vgaua6$11df5$1@dont-email.me>
 <8671079a0207b0f016ab1455868167011f3e65be@i2pn2.org>
 <vgcm06$1en9g$1@dont-email.me>
 <efd703ab2977082715e2adef7e49c18910c07d76@i2pn2.org>
 <vgd3ck$1grkg$1@dont-email.me>
 <fb5d2ffb843200379d4201b8a375e6fafa82ce48@i2pn2.org>
 <vggc5f$25spe$9@dont-email.me>
 <bbcc6772a61345be8f72179b65d3947016e06b9a@i2pn2.org>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Thu, 07 Nov 2024 10:28:16 +0100 (CET)
Injection-Info: dont-email.me; posting-host="a64f40a529641b77af8a405524a5b42f";
	logging-data="2723728"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/zHUIJQt763GIsNeF6hEbWcqriCcwcmvU="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:mjMmtfhiiJdTdtpw8arEFrNMAls=
Content-Language: en-US
In-Reply-To: <bbcc6772a61345be8f72179b65d3947016e06b9a@i2pn2.org>
Bytes: 2539

On 07.11.2024 01:51, Richard Damon wrote:
> On 11/6/24 1:22 PM, WM wrote:


>>>> These other intervals also have irrational endpoints. Every point 
>>>> outside of an interval is next to some endpoint which is irrational.
>>>
>>> Or rational endpoints.
>>
>> No, the rationals are centres of their intervals.
> 
> They can also be endpoints of intervals.

Not of their own intervals.
> 
>>>
>>> There is no "next to" on the dense line
>>
>> Every positive point is nearer to zero than to any negative point.
>> Of -x and 0 the latter is next to any positive x.
>>
> 
> But that isn't "next to".

It is next to when between a point and the interval no further point 
exists, like here:

Use the intervals J(n) = [n - 1/10, n + 1/10]. Without splitting or 
modifying them they can be translated and reordered, to cover the whole 
positive axis and every rational as a midpoint - if Cantor was right.

Regards, WM