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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 10 Jan 2025 07:41:30 -0500
Organization: i2pn2 (i2pn.org)
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On 1/9/25 5:29 PM, WM wrote:
> On 09.01.2025 22:06, joes wrote:
>> Am Thu, 09 Jan 2025 10:38:44 +0100 schrieb WM:
>>> On 09.01.2025 00:45, joes wrote:
>>>> Am Wed, 08 Jan 2025 23:06:27 +0100 schrieb WM:
>>>
>>>>> 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.
>>>> You can't talk about size without using |abs|.
>>> I can and I do. And everybody understands it in case of subsets. This
>>> proves, in this special case (and more is not required), that Cantor's
>>> size is only a qualitative measure, not a quantitative one.
>> You have not defined any other concept of "size".
> 
> I have in some cases. But even if had not,  cardinality would be unsharp 
> till useless since almost all sets have the same cardinality.
> 
> Regards, WM
> 

No, there are an infinite number of different sizes of sets.

The fact that all sets of size Aleph_0 are the same size isn't 
"unsharpness" but a precission that you don't understand. We can 
distinguish between the sizes of the sets like: The Set of All Natural 
Numbers, The Set of All Real Numbers, and The Set of all Real Functions.

THe fact that all countable infinite sets, like the Natural Numbers and 
the Rational Numbers are the same sizd, is just a fact, like that 1 == 
2/2, and your wanting them to be different is just showing your ignorance,