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

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
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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.