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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
Date: Wed, 27 Nov 2024 07:32:01 -0500
Organization: i2pn2 (i2pn.org)
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On 11/27/24 5:12 AM, WM wrote:
> On 26.11.2024 18:50, Richard Damon wrote:
>> On 11/26/24 11:59 AM, WM wrote:
>>> On 26.11.2024 16:06, Richard Damon wrote:
>>>> On 11/26/24 6:24 AM, WM wrote:
>>>>> On 26.11.2024 12:15, FromTheRafters wrote:
>>>>>> WM pretended :
>>>>>
>>>>>>> It is impossible to change |ℕ| by 1 or more.
>>>>>>
>>>>>> Right, sets don't change. The set {2,3,4,...} does not equal the 
>>>>>> set of natural numbers, but |{2,3,4,...}| does equal |N|.
>>>>>>
>>>>> Then your |N| is an imprecise measure. My |N| is precise.
>>>>> |{2,3,4,...}| = |N| - 1 =/= |N| .
>>>
>>>> Then your measure is incorrect, as by the DEFINITION of measures of 
>>>> infinite sets, all countably infinite sets have the same "measure".
>>>
>>> That is one special definition of a very imprecise measure. We can do 
>>> better.
>>
>> But maybe you can't and get something consistant.
> 
> Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1 is 
> consistent.
> 
> Regards, WM
> 
> 

So you think, but that is because you brain has been exploded by the 
contradiction.

We can get to your second set two ways, and the set itself can't know which.

We could have built the set by the operation of removing 1 like your 
math implies, or we can get to it by the operation of increasing each 
element by its successor, which must have the same number of elements, 
so we prove that in your logic |ℕ| - 1 == |ℕ|, which is one of Cantor's 
claim, and what you want to refute, but comes out of your "logic"

So saying that |ℕ| -1 is different than |ℕ| just makes your logic 
inconsistant.