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

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:
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What about {1, 2, 3, ..., n}, where n is taken to infinity? No limit?

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It's slightly complicated. :-P
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If we explicitly refer to sets, say, the sets S_1, S_2, S_3, ...
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We may call the sequence (S_1, S_2, S_3, ...) a "set sequence".
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Moreover we may define a certain limit (for such sequences) called "set limit".
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Then the following can be shown:
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      lim_(n->oo) {1, 2, 3, ..., n} = {1, 2, 3, ...} .
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Or, using defined symbols:
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      lim_(n->oo) F(n) = IN .
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[ The sequence here is (F(1), F(2), F(3), ...). It's limit IN. ]
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On the other hand:
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      lim_(n->oo) {n, n+1, n+2, ...} = {} .
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Hope this helps. :-P
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 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
?