Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

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.