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

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 = {{}, {{}}, ...

>
Typo, needs another closing curly bracket.
>

=>    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.

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!

--
Ben.