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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 22:01:27 -0800
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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.