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

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), ...). Its limit IN. ]
On the other hand:
      lim_(n->oo) {n, n+1, n+2, ...} = {} .
Hope this helps. :-P
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