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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: Tue, 19 Nov 2024 11:42:12 -0500
Organization: Peripheral Visions
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WM submitted this idea :
> 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.

You are not making any sense.