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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 23:01:00 +0100
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On 23.11.2024 22:48, Jim Burns wrote:
> On 11/23/2024 3:45 PM, WM wrote:

>>> ⎜ Assume that there are
>>> ⎜ enough red hats for the first 𝔊 numbers
>>> ⎜ but not enough for the 𝔊+1ᵗʰ
>>
>> That is a mistake.
>> If there are enough hats for G natnumbers,
>> then there are also enough for G^G^G natnumbers.
> 
> Thank you.
> 
>> Alas they leave G^G^G unit intervals without hats.
>> That is the catch!
> 
> After all hat.shifts,
> there is no first number without a hat.

But almost all numbers are without hat because the number of hats has 
not increased.

> After all hat.shifts,
> the set of numbers without hats is empty.

Then the number of hats must have increased.
Then number theory can be wasted, because its formula for n/logn primes 
in (0, n] is no longer valid after Cantor-reordering.
> 
>>>> There are too few prime numbers.
>>>
>>> No,
>>> there being too few primes
>>> leads to contradiction.
>>
>> Shall unit intervals disappear like Bob?
> 
> After all swaps,
> Bob is not in any room (visible or dark)

He is in a dark room.

> which Bob has ever been in.

But the unit intervals remain steadfast on the real line.

Regards, WM