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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 24 Nov 2024 21:56:54 +0100
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On 24.11.2024 21:17, Jim Burns wrote:
> On 11/24/2024 2:42 PM, WM wrote:
>> On 24.11.2024 20:26, Jim Burns wrote:
> 
>>> What we mean by
>>>   |E(k)| ≤ |E(k+1)|
>>> is that
>>> there is a one.to.one function
>>>   from E(k) to E(k+1)
>>> The successor operation, for example.
>>
>> What I mean is the fact that
>> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
>> whereas Cantor's ℵo is a very unsharp measure.
> 
> Finite cardinalities can change by 1.

Endsegmentes can change by 1 element. Therefore their number of elements 
can change by 1.
> 
> Infinite cardinalities are larger than
> each finite cardinality,
> and cannot change by 1.

Infinite cardinalities are to coarse to indicate that change. But the 
change takes place:
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
==> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1

> ℕ is the set of each and only finite cardinalities.
> 
> |ℕ| isn't a finite cardinality.
> |ℕ| cannot change by 1.

ℕ cannot change by 1 Element.
==> |ℕ| can change by 1.

>> Cantor's nonsense has many faces.
>> It i not suitable for serious maths.
> 
> Cardinalities which cannot change by 1
> do not change by 1 when they're called unserious.

 From two examples above you can see a change of an infinite set by 1 
element. A proper measure will be able to indicate that.

Why do you wish to adhere to such an incapable measure?

Regards, WM