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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Tue, 19 Nov 2024 16:24:52 +0100
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On 18.11.2024 23:40, FromTheRafters wrote:
> 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.

Why? Because only potential infinity is involved. True bijections pr5ove 
equinumerosity.
>>>> 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!

The number of ℕ \ {1} is 1 less than ℕ.
> 
>>>> 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.

It is, because 4 and 8 are missing.

 > There is a bijection.

Only between numbers which have more successors than predecessors, 
although it is claimed that no successors are remaining.

Regards, WM

Regards, WM