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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: Mon, 18 Nov 2024 17:35:42 +0100
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On 17.11.2024 21:51, FromTheRafters wrote:
> It happens that WM formulated :
>> On 17.11.2024 17:59, FromTheRafters wrote:
>>> WM pretended :
>>>> On 17.11.2024 12:38, FromTheRafters wrote:
>>>>> WM presented the following explanation :
>>>>>> On 17.11.2024 12:01, FromTheRafters wrote:
>>>>>>> WM was thinking very hard :
>>>>>>>> On 16.11.2024 22:33, Moebius wrote:
>>>>>>>>
>>>>>>>>> For example "aleph_0 - aleph_0" is not defined.
>>>>>>>>
>>>>>>>> Small wonder. ℵo means only infinitely many: |ℕ|, |ℚ|, and many 
>>>>>>>> others.
>>>>>>>> |ℕ|-|ℕ| however is defined.
>>>>>>>
>>>>>>> No, it is not.
>>>>>>
>>>>>> If sets are invariable then ℕ \ ℕ is empty.
>>>>>> If |ℕ| concerns only the elements of ℕ, then |ℕ|-|ℕ|= 0.
>>>>>
>>>>> So, you're saying that if I take aleph_zero natural numbers and I 
>>>>> remove the aleph_zero odd numbers from consideration in a new set, 
>>>>> I will have a new emptyset instead of E?
>>>>
>>>> Try to understand. "aleph_0 - aleph_0" is not defined.
>>>
>>> Try to understand that |N| equals aleph_zero.
>>
>> Of course. ℵo equals |ℕ|, equals |ℚ|, equals all countable sets. It is 
>> simply another name for infinitely "many". |ℕ| however is a fixed 
>> infinite number. Note that sets are invariable.
> 
> But you said both that it equals zero and that it is undefined. You 
> should pick one and be consistent.

I said that |ℕ| and |ℚ| and |ℕ| - 5 and |ℕ| + 6 etc. equal ℵo. That 
means that ℵo is nothing but "infinitely many". Therefore ℵo - ℵo is 
undefined but |ℕ| - |ℕ| = 0 and |ℚ| - |ℕ| > 0.

|ℕ| - |ℕ| = 0 because if you subtract one element from ℕ then you have 
no longer ℕ and therefore no longer |ℕ| describing it.

Regards, WM