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

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

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).

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

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Nope, n may be any natural number [actually, "n" is a variable here].

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

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Right.

Thank you Moebius. :^)

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So, if n = all_of_the_naturals, then

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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.

Indeed! *sigh*

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

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

Right?

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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, ...

aka ({0}, {0, 1}, {0. 1, 2}, ...) with 0 := {}.

does have a set-theoretical limit: N.

Ben Bacarisse is -as always- quite right here.

However, there is no largest natural number, when I think of that I see no limit to the naturals. I must be missing something here? ;^o

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