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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Thu, 21 Nov 2024 08:05:38 -0500
Organization: Peripheral Visions
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WM wrote :
> On 21.11.2024 00:01, FromTheRafters wrote:
>> After serious thinking WM wrote :
>>> On 20.11.2024 21:05, FromTheRafters wrote:
>>>> WM wrote on 11/20/2024 :
>>>>> On 20.11.2024 15:14, FromTheRafters wrote:
>>>>>> WM formulated on Wednesday :
>>>>>
>>>>>>>
>>>>>>> It does not make it wrong, but it unmasks it at imprecise. That's why 
>>>>>>> I don't like it. We can do better.
>>>>>>
>>>>>> It works well enough.
>>>>>
>>>>> Really? Then you can answer the following questions:
>>>>>
>>>>> Let every unit interval after a natural number on the real axis be 
>>>>> coloured white with exception of the powers of 2 which are coloured 
>>>>> black. Is it possible to shift the black intervals so that the whole 
>>>>> real axis becomes black?
>>>>
>>>> No, of course not.
>>>
>>> Thank you. You are the first person not bewitched by set theory.
>> 
>> The real line is infinitely long in both directions. Your shifted intervals 
>> are all on the positive real axis.
>
> But if Cantor was right, all integers, positive as well as all other could be 
> in bijection with the powers of 2.

Integer powers of two? Yes, both are countable sets.

>>>> Depending on the order of the tasks. I think half red or half black.
>>>
>>> Recidivistic?
>> 
>> I recognize McDuck in disguise, or your notion that there are fewer primes 
>> than there are natural numbers. You keep finding different ways to say the 
>> same wrong thing.
>
> The analytical limit is wrong in your opinion?

Which limit? Is a half-plane smaller than a whole plane? Is a ray 
shorter than a line? Is a line segment shorter than a ray? What size 
symbol could you write on a whole plane that you could not write on a 
half plane? If McDuck's earnings always outpace his spending how can 
one consider him ever going bankrupt 'in the limit'? Which limit?