Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 3 Dec 2024 22:54:27 +0100
Organization: A noiseless patient Spider
Lines: 85
Message-ID: <vinumj$c7p5$3@dont-email.me>
References: <vg7cp8$9jka$1@dont-email.me>
 <71758f338eb239b7419418f49dfd8177c59d778b@i2pn2.org>
 <via83s$jk72$2@dont-email.me> <viag8h$lvep$1@dont-email.me>
 <viaj9q$l91n$1@dont-email.me> <vibvfo$10t7o$1@dont-email.me>
 <vic6m9$11mrq$4@dont-email.me> <vicbp2$1316h$1@dont-email.me>
 <vid4ts$1777k$2@dont-email.me> <vidcv3$18pdu$1@dont-email.me>
 <bdbc0e3d-1db2-4d6a-9f71-368d36d96b40@tha.de> <vier32$1madr$1@dont-email.me>
 <vierv5$1l1ot$2@dont-email.me> <viiqfd$2qq41$5@dont-email.me>
 <vijhrd$34mp8$1@dont-email.me> <vilh59$3k21l$5@dont-email.me>
 <vilheq$3ks01$3@dont-email.me> <vilhjk$3k21l$9@dont-email.me>
 <vilhk8$3ks01$4@dont-email.me> <vilhnl$3k21l$10@dont-email.me>
 <viljdo$3k21l$12@dont-email.me> <vim52i$3s6qb$1@dont-email.me>
 <vim58c$3sjje$1@dont-email.me> <vim61c$3s6qb$3@dont-email.me>
 <vim6rm$3sjje$3@dont-email.me> <vim76d$3sjje$4@dont-email.me>
 <vim7p5$3t271$1@dont-email.me> <vim86l$3s6qb$9@dont-email.me>
 <vimfgq$3ui70$3@dont-email.me> <vinrql$b2tq$1@dont-email.me>
Reply-To: invalid@example.invalid
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Tue, 03 Dec 2024 22:54:28 +0100 (CET)
Injection-Info: dont-email.me; posting-host="5c7b806b0e666b2f903894d226c0c641";
	logging-data="401189"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18FskeePSMlqidvbAg6XtyV"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:T9MfxJLs6y2Rykfi0YhE/E1pxZk=
Content-Language: de-DE
In-Reply-To: <vinrql$b2tq$1@dont-email.me>
Bytes: 4967

Am 03.12.2024 um 22:05 schrieb Chris M. Thomasson:
> On 12/3/2024 12:29 AM, Moebius wrote:
>> Am 03.12.2024 um 07:24 schrieb Moebius:
>>> Am 03.12.2024 um 07:17 schrieb Chris M. Thomasson:
>>>
>>>>>>
>>>>>>              0
>>>>>>             / \
>>>>>>            /   \
>>>>>>           /     \
>>>>>>          /       \
>>>>>>         1         2
>>>>>>        / \       / \
>>>>>>       /   \     /   \
>>>>>>      3     4   5     6
>>>>>>   .........................
>>>
>>> Though we may take 1 for the root too. This way we would get (using 
>>> binary representation):
>>>
>>>  >>>              1
>>>  >>>             / \
>>>  >>>            /   \
>>>  >>>           /     \
>>>  >>>          /       \
>>>  >>>        10        11
>>>  >>>        / \       / \
>>>  >>>       /   \     /   \
>>>  >>>     100   101 110   111
>>>  >>>   .........................
>>>
>>> I guess you get the pattern. :-P
>>
>> Another one (without numbers) but /left/ (l), /right/ (r):
>>
>>>  >>>              *
>>>  >>>             / \
>>>  >>>            /   \
>>>  >>>           /     \
>>>  >>>          /       \
>>>  >>>         l         r
>>>  >>>        / \       / \
>>>  >>>       /   \     /   \
>>>  >>>     ll    lr   rl    rr
>>>  >>>   .........................
>>
>> This way we may even "identify" each node in the tree with a (finite) 
>> l- r-sequence:
>>
>>>  >>>             ()       [<<< the "empty l-r-sequence"]
>>>  >>>            /  \
>>>  >>>           /    \
>>>  >>>          /      \
>>>  >>>         /        \
>>>  >>>       (l)        (r)
>>>  >>>       / \        / \
>>>  >>>      /   \      /   \
>>>  >>>   (l,l) (l,r) (r,l) (r,r)
>>>  >>>   .........................
>>
>> :-P
>>
>> So each node actually "is" (or represents) the path leading to it. :-P
> 
> (l, l) means take two left branches from the root.
> 
> (r, l) means take one right and one left from the root.
> 
> I see the pattern. It makes me think some more about my original tree 
> with node, say, 6.
> 
> It's parent is (6-2)/2 = 2 that is a right (-2) wrt 6 has a parent at 2.
> 
> At node 2 take a right to get at node 6
> 
> Now, lets try 5. It's parent is (5-1)/2 = 2. That is a left from 2 wrt 
> (-1). There is a pattern here as well.
> 
> At node 2 take a left to get at node 5.
> 
> Make any sense to you? Thanks.

I'm sorry, no time for deeper analysis. :-/