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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Mon, 2 Dec 2024 22:07:09 -0800
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On 12/2/2024 10:01 PM, Chris M. Thomasson wrote:
> On 12/2/2024 9:47 PM, Moebius wrote:
>> Am 03.12.2024 um 06:34 schrieb Chris M. Thomasson:
>>
>>> What about {1, 2, 3, ..., n}, where n is taken to infinity? No limit?
>>
>> It's slightly complicated. :-P
>>
>> If we explicitly refer to sets, say, the sets S_1, S_2, S_3, ...
>>
>> We may call the sequence (S_1, S_2, S_3, ...) a "set sequence".
>>
>> Moreover we may define a certain limit (for such sequences) called 
>> "set limit".
>>
>> Then the following can be shown:
>>
>>       lim_(n->oo) {1, 2, 3, ..., n} = {1, 2, 3, ...} .
>>
>> Or, using defined symbols:
>>
>>       lim_(n->oo) F(n) = IN .
>>
>> [ The sequence here is (F(1), F(2), F(3), ...). It's limit IN. ]
>>
>> On the other hand:
>>
>>       lim_(n->oo) {n, n+1, n+2, ...} = {} .
>>
>> Hope this helps. :-P
>>
>> .
>> .
>> .
>>
> 
> Sometimes I like to think of the set of all natural numbers as an n-ary 
> tree, binary here, wrt zero as a main root, so to speak:
> 
>              0
>             / \
>            /   \
>           /     \
>          /       \
>         1         2
>        / \       / \
>       /   \     /   \
>      3     4   5     6
>   .........................
> 
> On and on. A lot of math can be applied to it.

wrt 2-ary:

Roots of 5 and 6:
(5-1)/2 = 2
(6-2)/2 = 2

Roots of 3 and 4:
(3-1)/2 = 1
(4-2)/2 = 1

Roots of 1 and 2:
(1-1)/2 = 0
(2-2)/2 = 0

Roots of 7 and 8:
(7-1)/2 = 3
(8-2)/2 = 3

?