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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: Tue, 26 Nov 2024 06:19:03 -0500
Organization: Peripheral Visions
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WM laid this down on his screen :
> On 25.11.2024 19:08, Jim Burns wrote:
>> On 11/25/2024 7:11 AM, WM wrote:
>>> On 24.11.2024 19:16, Jim Burns wrote:
>>>> On 11/24/2024 6:06 AM, WM wrote:
>>>>> On 24.11.2024 06:37, Jim Burns wrote:
>> 
>>>>>> Consider the set of natural numbers for which
>>>>>> there are NOT enough hats.
>>>>>
>>>>> It is dark.
>>>>
>>>> It is not dark what we mean by 'natural number'.
>>>> A natural number is countable.to from.0
>>>
>>> That is a definable number.
>> 
>> Thus
>> it is not dark that
>> what we mean by 'natural number' is
>> countable.to from.0  and is
>> what you call a definable number.
>
> Consider the black hats at every 10 n and white hats at all other numbers n. 
> It is possible to shift the black hats such that every interval (0, n] is 
> completely covered by black hats. There is no first n discernible that cannot 
> be covered by  black hat. But the origin of each used black hat larger than n 
> is now covered by a white hat. Without deleting all white hats it is not 
> possible to cover all n by black hats. But deleting white hats is prohibited 
> by logic. Exchanging can never delete one of the exchanged elements. 
> Therefore we have here, like in all Cantor-pairings, the same impediment and 
> further disussion is futile: You must deny logic. I do not.

How much paint would be needed to paint the entire cartesian plane? Too 
much huh? then just paint the first quadrant with one quarter the 
amount of paint. Your 'logic' is too easily denied.