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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Mon, 18 Nov 2024 17:40:03 -0500
Organization: Peripheral Visions
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WM wrote on 11/18/2024 :
> On 18.11.2024 22:58, FromTheRafters wrote:
>> on 11/18/2024, WM supposed :
>>> On 18.11.2024 18:15, FromTheRafters wrote:
>>>> WM brought next idea :
>>>
>>>>>> |ℕ| - |ℕ| = 0 because if you subtract one element from ℕ then you have
>>>>> no longer ℕ and therefore no longer |ℕ| describing it.
>>>>
>>>> Still wrong.
>>>
>>> If you remove one element from ℕ, then you have still ℵo but no longer all 
>>> elements of ℕ.
>> 
>> But you do have now a proper subset of the naturals the same size as 
>> before.
>
> It has one element less, hence the "size" ℵo is a very unsharp measure.

Comparing the size of sets by bijection. Bijection of finite sets give 
you a same number of elements, bijection of infinite sets give you same 
size of set.

>>> If |ℕ| describes the number of elements, then it has changed to |ℕ| - 1.
>> 
>> Minus one is not defined.
>
> Subtracting an element is defined. |ℕ| - 1 is defined as the number of 
> elements minus 1.

Nope!

>>> If you don't like |ℕ| then call this number the number of natural numbers.
>> 
>> Why would I do that when it is the *SIZE* of the smallest infinite set.
>
> The set of prime numbers is smaller.

No, it is not. There is a bijection.