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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 5 Jan 2025 14:22:56 -0800
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On 1/5/2025 3:07 AM, WM wrote:
> On 04.01.2025 17:20, Jim Burns wrote:
>> On 1/4/2025 3:42 AM, WM wrote:
>>> On 1/3/2025 3:56 PM, Jim Burns wrote:
>>
>>>> All finite.ordinals removed from
>>>> the set of each and only finite.ordinals
>>>> leaves the empty set.
>>>
>>> But removing
>>> every ordinal that you can define
>>> (and all its predecessors) from ℕ leaves
>>> almost all ordinals in ℕ.
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>
>> ℕ is the set of each and only finite.ordinals.
> 
> Yes.
>> |ℕ| := ℵ₀ = |ℕ\{0}| = |ℕ\{0,1}| = ... =
>> |ℕ\{0,1,...,n}| = ...
>>
>> The sequence of end.segments of ℕ
>> grows emptier.one.by.one but
>> it doesn't grow smaller.one.by.one.
> 
> It does but you cannot give the numbers because they are dark.

Here is a dark number for ya, when I realize that you are a teacher...

666

?



> A precise measure must detect the loss of one element. ℵo is no precise 
> measure but only another expression for infinitely many.
>>
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>
>> ℕ is the set of each and only finite.ordinals.
> 
> Yes. But most of them cannot be named as individuals and then removed 
> because ℵo will always remain in the set. Collectively however is works 
> ℕ \ {1, 2, 3, ...} = { }.
>>
>> Each finite.ordinal is not weird.
>> Even an absurdly.large one like Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
>> is not weird.
> 
> Numbers which can be individualized are far less than 1 % of |ℕ|
> 
> Regards, WM
>