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

Ben Bacarisse wrote on 12/4/2024 :

Moebius <invalid@example.invalid> writes:
>

Am 04.12.2024 um 12:26 schrieb Ben Bacarisse:

FromTheRafters <FTR@nomail.afraid.org> writes:
 

Moebius expressed precisely :

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.).

 If you say so, but I haven't seen this written anywhere.

 @FromTheAfter: https://en.wikipedia.org/wiki/Set-theoretic_limit
 

It's usually framed in terms of least upper bounds, so that might be why
you are not recalling it.
Ironically, there is a very common example of a "set theoretic limit"
which is the point-wise limit of a sequence of functions.  Since
functions are just sets of pairs, these long-known limits are just the
limits of sequences of sets.  It's ironic because WM categorically
denies that /any/ non-constant sequence of sets has a limit, yet the
basic mathematics textbook he wrote includes the definition of the
point-wise limit, as well as stating that functions are just sets of
pairs.  He includes examples of something he categorically denies!

 Same with the notions of /bijections/. Explained in his book but denied by
WM these days.

>
Ah, I had not seen him deny the notion of a bijection.  Do you have a
message ID?  I used to collect explicit statements, though I don't post
enough to make it really worth while anymore.

It fits in with his general idea that his dark numbers cannot be in a bijection or Cantor Pairing because they are dark. He refutes Cantor.