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

On 12/4/2024 6:11 AM, Ben Bacarisse wrote:

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
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        (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.
 

I think he has said that Cantor Pairing does not work with "certain" natural numbers? I guess, the dark ones? Afaict, Cantor Pairing works with any natural number and can generate a _unique_ pair of natural numbers. The pair can also be mapped back into the single natural that created it to begin with. Nothing is lost. The pairing works fine.
Not sure why WM thinks that Cantor Pairing does not work with any natural number... I think I am not misunderstanding WM here.