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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: Wed, 27 Nov 2024 20:25:45 +0100
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On 27.11.2024 19:19, FromTheRafters wrote:
> WM explained :
>> On 27.11.2024 13:32, Richard Damon wrote:
>>> On 11/27/24 5:12 AM, WM wrote:
>>
>>>> Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1 
>>>> is consistent.
>>>
>>> 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,
>>
>> Yes, the same number of elements, but not the same number of natural 
>> numbers.
>>
>> Hint: Decreasing every element in the real interval (0, 1] by one 
>> point yields the real interval [0, 1). The set of points remains the 
>> same, the set of positive points decreases by 1.
> 
> If you have a successor function for the real numbers, why don't you 
> share it with the rest of the world?

I don't because almost all real numbers are dark. Nevertheless we know 
that the interval (0, 1] contains all small positive numbers. Hence 
shifting it by 1 point we get [0, 1).

Regards, WM