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

Am 03.12.2024 um 01:29 schrieb Chris M. Thomasson:

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:
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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 here may be any element in IN.

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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.

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Thank you Moebius. :^)

 So, if n = all_of_the_naturals, then

No, /n/ has to be a natural number, i.e. an element in IN.
Check the DEFINITION again:
| Def.: F(n) =df {1, 2, 3, ..., n}    (n e IN).
"(n e IN)" means/states that only terms which refer to natural numbers are allowed here. [Thoug you may use other symbols instead of "n".]
Hint: "all_of_the_naturals" does not denote a natural number.
But the terms "1", "2", etc. ... do. :-P
You may think of "(n e IN)" as some type information concerning the (defined) expression"F(n)". :-)

{ 1, 2, 3, ... }

That's just a common symbol to denote IN (the set of natural numbers).
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Hm... Maybe your problem here is that NEITHER the symbol "{1, 2, 3, ..., n}" NOR the symbol "{1, 2, 3, ...}" have been (formally) defined.
So let's do it now! (Better late than never.)
| Def. {1, 2, 3, ..., n} =df {m e IN : m <= n}        (n e IN)
| Def. {1, 2, 3, ...} =df IN .
.
.
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