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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: Wed, 8 Jan 2025 23:06:27 +0100
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On 08.01.2025 20:19, Jim Burns wrote:
> On 1/8/2025 4:16 AM, WM wrote:
>> On 08.01.2025 00:50, Jim Burns wrote:
> 
>>> The cardinal:ordinal distinction
>>> -- which does not matter in the finite domain
>>> matters in the infinite domain.
>>
>> The reason is that
>> the infinite cardinal ℵ₀ is based on
>> the mapping of the potentially infinite collection of
>> natural numbers n,
>> all of which have
>> infinitely many successors.
>> The cardinal ℵ₀ is not based on
>> the mapping of
>> the actually infinite set ℕ where
>> ℕ \ {1, 2, 3, ...} = { }.
> 
> For each set smaller.than a fuller.by.one set,
> the cardinal:ordinal distinction doesn't matter.
> Cardinals and ordinals always go together.
> 
> For each set smaller.than a fuller.by.one set
> there is an ordinal of its size in
> the set ℕ of all finite ordinals.
> 
> Each set for which
> there is NOT an ordinal of its size in
> the set ℕ of all finite ordinals
> is NOT a set smaller.than a fuller.by.one set.

The set {1, 2, 3, ...} is smaller by one element than the set {0, 1, 2, 
3, ...}. Proof: {0, 1, 2, 3, ...} \ {1, 2, 3, ...} = {0}. Cardinality 
cannot describe this difference because it covers only mappings of 
elements which have almost all elements as successors.

Regards, WM