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

Am 04.12.2024 um 02:02 schrieb Moebius:

Am 04.12.2024 um 01:47 schrieb Chris M. Thomasson:

On 12/3/2024 2:32 PM, Moebius wrote:

Am 03.12.2024 um 23:16 schrieb Moebius:

Am 03.12.2024 um 22:59 schrieb Chris M. Thomasson:

>

However, there is no largest natural number, when I think of that I see no limit to the naturals.

>
Right. No "coventional" limit. Actually,
>
      "lim_(n->oo) n"
>
does not exist.

>
In the sense of as n tends to infinity there is no limit that can be reached [...]?

 Exactly.
 We say, n is "growing beyond all bounds". :-P

On the other hand, if we focus on the fact that the natural numbers are sets _in the context of set theory_, namely
       0 = {}, 1 = {{}}, 2 = {{}, {{}}, ...
=>    0 = {}, 1 = {0}, 2 = {0, 1}, ...
(due to von Neumann)
then we may conisider the "set-theoretic limit" of the sequence
      (0, 1, 2, ...) = ({}, {0}, {0, 1}, ...).
This way we get:
      LIM_(n->oo) n = {0, 1, 2, ...} = IN. :-P
I'd like to mention that "lim_(n->oo) n" is "old math" (oldies but goldies) while "LIM_(n->oo) n" is "new math" (only possible after the invention of set theory (->Cantor) and later developments (->axiomatic set theory, natural numbers due to von Neumann, etc.).