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

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

On 12/4/2024 2:56 AM, Ben Bacarisse wrote:

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
 

On 12/3/2024 3:35 AM, Ben Bacarisse wrote:

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>

On 12/2/2024 4:00 PM, Chris M. Thomasson wrote:

On 12/2/2024 3:59 PM, Moebius wrote:

Am 03.12.2024 um 00:58 schrieb Chris M. Thomasson:

On 12/2/2024 3:56 PM, Moebius wrote:

Am 03.12.2024 um 00:51 schrieb Chris M. Thomasson:

On 12/1/2024 9:50 PM, Moebius wrote:

Am 02.12.2024 um 00:11 schrieb Chris M. Thomasson:

On 11/30/2024 3:12 AM, WM wrote:

>

Finite initial segment[s]: F(n) = {1, 2, 3, ..., n}    (n e IN).

[...]
>
When WM writes:
>
{1, 2, 3, ..., n}
>
I think he might mean that n is somehow a largest natural number?

>
Nope, he just means some n e IN.

>
So if n = 5, the FISON is:
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{ 1, 2, 3, 4, 5 }
>
n = 3
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{ 1, 2, 3 }
>
Right?

>
Right.

Thank you Moebius. :^)

>
So, i n = all_of_the_naturals, then

You are in danger of falling into one of WM's traps here.  Above, you
had n = 3 and n = 5.  3 and 5 are naturals.  Switching to n =
all_of_the_naturals is something else.  It's not wrong because there are
models of the naturals in which they are all sets, but it's open to
confusing interpretations and being unclear about definition is the key
to WM's endless posts.
>

{ 1, 2, 3, ... }
>
Aka, there is no largest natural number and they are not limited. Aka, no
limit?

The sequence of FISONs has a limit.  Indeed that's one way to define N
as the least upper bound of the sequence
   {1}, {1, 2}, {1, 2, 3}, ...
although the all terms involved need to be carefully defined.
>

Right?

The numerical sequence 1, 2, 3, ... has no conventional numerical limit,
but, again, if the symbols 1, 2, 3 etc stand for sets (as in, say, Von
Neumann's model for the naturals) then the set sequence
   1, 2, 3, ...
does have a set-theoretical limit: N.

>
However, there is no largest natural number,

Yes, there is no largest natural.  Let's not loose sight of that.
 

when I think of that I see no
limit to the naturals. I must be missing something here? ;^o

It's just that there are lots of kinds of limit, and a limit is not
always in the set in question.  Very often, limits take us outside of
the set in question.  R (the reals) can be defined as the "smallest" set
closed under the taking of certain limits -- the limits of Cauchy
sequences, the elements of which are simply rationals.

>
So, the limit of the natural numbers is _outside_ of the set of all natural
numbers?

Yes.

Even if we don't consider FISONs, we can define a limit (technically a
least upper bound) for the sequence 1, 2, 3, ...  It won't be a natural
number.  We will have to expand our ideas of "number" and "size" to get
the smallest "thing", larger than all naturals.  This is how the study
of infinite ordinals starts.
 

>
Okay... I need to ponder on that. Can I just, sort of, make up a "symbol"
and say this is larger than any natural number, however it is definitely
_not_ a natural number in and of itself?

Yes, but everyone else already uses the symbol omega for it, so don't
make up a new one.

Damn. Still missing something here. Thanks for your patience... ;^o

I don't think you are missing much in the big picture.  The details are
hairy but you can find them in a textbook on set theory.

--
Ben.