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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: Fri, 27 Dec 2024 11:14:37 +0100
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On 26.12.2024 19:41, Jim Burns wrote:
> On 12/22/2024 6:32 AM, WM wrote:

> I (JB) think that it may be that
> 'almost.all' '(∀)' refers concisely to
> the differences in definition at
> the center of our discussion.
> 
> For each finite.cardinal,
> almost.all finite.cardinals are larger.
> ∀j ∈ ℕⁿᵒᵗᐧᵂᴹ: (∀)k ∈ ℕⁿᵒᵗᐧᵂᴹ: j < k

That is true in potential infinity. It is wrong in actual infinity 
because there ℕ \ {1, 2, 3, ...} = { } shows that all finite cardinals 
can be manipulated such that none is larger.
> 
> I think that you (WM) would deny that.
> You would say, instead,
> ᵂᴹ⎛ for each definable finite.cardinal
> ᵂᴹ⎜ almost.all finite cardinals are larger.

Right.
> For sequence ⟨Sₙ⟩ₙ᳹₌₀ of sets
> Sₗᵢₘ is a limit.set of ⟨Sₙ⟩ₙ᳹₌₀
> if
> each x ∈ Sₗᵢₘ is ∈ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀  and
> each y ∉ Sₗᵢₘ is ∉ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀
> ⟨Sₙ⟩ₙ᳹₌₀ ⟶ Sₗᵢₘ  ⇐
> ⎛ x ∈ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  x ∈ Sₖ
> ⎝ y ∉ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  y ∉ Sₖ
> 
A limit is a set S​͚  such that nothing fits between it and all sets of 
the sequence.
> ---
> The notation a​͚  or S​͚  for aₗᵢₘ or Sₗᵢₘ
> is tempting, but
> it gives the unfortunate impression that
> a​͚  and S​͚  are the infinitieth entries of
> their respective infinite.sequences.
> They aren't infinitieth entries.
> They are defined differently.

The last natural number is finite, and therefore objectively belongs to 
a finite set. But like all dark numbers it has no FISON and therefore 
the dark realm appears like an infinite set.
> 
>>> E(n+2) is
>>> the set of all finite.cardinals > n+2
>>> E(n+1)  =  E(n+2)∪{n+2}
>>> E(n+2)∪{n+2} isn't larger.than E(n+2)
>>
>> Wrong.
> 
> Almost all finite.cardinals are larger than
> finite.cardinal n+1
> {n+2} isn't large enough to change that.
> Almost all finite.cardinals are larger than
> finite cardinal n+2.

That is true for visible n.
> 
>>> #E(n+2) isn't any of the finite.cardinals in ℕ
>>
>> It is an infinite number but
>> even infinite numbers differ like |ℕ| =/= |ℕ| + 1.
> 
> Infiniteᵂᴹ numbers which differ like |ℕ| ≠ |ℕ| + 1.
> are finiteⁿᵒᵗᐧᵂᴹ numbers.

No. They are invariable numbers like ω and ω+1. The alephs differ only 
because they count potentially infinite sets which always can be 
bijected as far as is desired.
> 
> The concept of 'limit' is a cornerstone of
> calculus and analysis and topology.

A limit is a number or set such that nothing fits between it and all 
numbers or sets of the sequence.
> 
> That cornerstone rests upon 'almost.all'.
> Each finite.cardinal has infinitely.more
> finite.cardinals after it than before it.

For each finite cardinal that can be defined this is true. Dark 
cardinals never have been considered.

> If one re.defines things away from that,
> it is only an odd coincidence if, after that,
> any part of calculus or analysis or topology
> continues to make sense.
> 
Without dark cardinals set theory does not make sense.
∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
and
|ℕ \ {1, 2, 3, ...}| = 0
would contradict each other because more than all n are not in {1, 2, 3, 
....}.

Regards, WM