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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: Sat, 23 Nov 2024 20:40:40 +0100
Organization: A noiseless patient Spider
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On 23.11.2024 19:43, FromTheRafters wrote:
> WM wrote on 11/23/2024 :
>> On 23.11.2024 13:35, FromTheRafters wrote:
>>> WM has brought this to us :
>>>> On 23.11.2024 13:20, FromTheRafters wrote:
>>>>> WM laid this down on his screen :
>>>>
>>>>>> Let every unit interval after a natural number on the real axis be 
>>>>>> coloured white with exception of the intervals after the prime 
>>>>>> numbers which are coloured red. It is impossible to shift the red 
>>>>>> intervals so that the whole real axis becomes red. Every interval 
>>>>>> (10n, 10 (n+1)] is deficient - on the whole real axis.
>>>>>
>>>>> So what? Your imaginings don't affect the fact that there is a 
>>>>> bijection.
>>>>
>>>> If there was a bijection,
>>>
>>> There is.
>>>
>>>> then the whole axis could become red.
>>>
>>> What makes you think that?
>>
>> A bijection proves that every prime number (and its colour) can be put 
>> to a natural number (and colour it).
> 
> ???

A bijection between natural numbers and prime numbers proves that for 
every prime number there is a natural number: p_1, p_2, p_3, ...
If that is correct, then there are as many natural numbers as prime 
numbers and as many prime numbers as natural numbers. Then the following 
scenario is possible:

Cover the unit intervals of prime numbers by red hats. Then shift the 
red hats so that all unit intervals of the positive real axis get red hats.

Regards, WM