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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 5 Jan 2025 12:07:17 +0100
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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.
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