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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: Sat, 23 Nov 2024 21:22:17 -0500
Organization: i2pn2 (i2pn.org)
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On 11/23/24 4:11 PM, WM wrote:
> On 23.11.2024 21:46, Richard Damon wrote:
>> On 11/23/24 2:40 PM, WM wrote:
>>> 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
>>>
>>
>> And you can, as
>> the red hat on the number 2, can be moved to the number 1
>> the red hat on the number 3, can be moved to the number 2
>> the red hat on the number 5, can be moved to the number 3
> 
> A very naive recipe.

But it works.

>>
>> and in general, the red hat on the nth prime number can be moved to 
>> the number n
>>
>> Since there are a countable infinite number of prime numbers, there 
>> exist an nth prime number for every n,
> 
> Yes, for every n that belongs to a tiny initial segment.

No, for EVERY n.

Show one that it doesn't work for!

> 
>> so all the numbers get covered.
> 
> No.

WHich one doesn't.

Your LYING claims just prove your ignorance.

>>
>> We have a 1:1 relationship (bijection) established between the set of 
>> prime numbers and the set of Natural Numbers.
> 
> No. Every hat taken from wherever leaves there a naked unit interval. 
> Therefore for every interval (0, n] inside which hats are moved, the 
> relative covering is about n/logn.

Yes, it leave a naked unit interval, that will later be replaced by the 
n-th prime.

Your problem is you apparently you can only think about finite sets, so 
you brain just explodes when you try to handle a truely infinite one.

Infinity isn't just a redeculously big number, it is a different class 
of numbers, something your "finite number" logic just doesn't handle.


> 
> Regards, WM