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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: Wed, 8 Jan 2025 14:29:30 -0800
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On 1/8/2025 2:06 PM, WM wrote:
> 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.

Huh? What about:

{ 1, 2, 3, ... } = { 1, 2, 3, ... }

{ 1 - 1, 2 - 1, 3 - 1, ... } = { 0, 1, 2, ... }

They both have the same number of elements. Infinite.