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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Sat, 23 Nov 2024 21:35:36 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Sat, 23 Nov 2024 21:45:06 +0100 schrieb WM:
> On 23.11.2024 21:18, Jim Burns wrote:
>> On 11/23/2024 5:30 AM, WM wrote:
>>> On 22.11.2024 22:50, Jim Burns wrote:
>> 
>>>> ℙ covers ℕ, and ℕ covers ℙ
>>>
>>> Let every unit interval on the infinite real axis be coloured white.
>>> Cover the unit intervals of prime numbers by red hats.
>>> It is impossible to shift the red hats
>> 
>> Yes, because we are finite beings,
>> and there are infinitely.many red hats.
> 
> No, the reason is that every shift removes the hat from its place and
> requires an other hat, taken from wherever, but with certainty leaving
> an uncovered interval. That does never change.
No. That other place where we take the hats from, larger primes, are of
course covered by even larger primes. We don't stop at some arbitrary
finite number, but continue forever.

>>> It is impossible to shift the red hats so that all unit intervals of
>>> the whole real axis get red hats.
>>> There are too few prime numbers.
>> No. Assume that that is so.
> That need not be assumed but that is obviously so for every part of the
> real axis.
Not true. The first n prime numbers are obviously in bijection with the
numbers 1 to n.

>> 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. Alas they leave G^G^G unit
> intervals without hats. That is the catch!
Right, that is your mistake. There are still hats left over.

>>> There are too few prime numbers.
>> No, there being too few primes leads to contradiction.
> Shall unit intervals disappear like Bob?
Like who?

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.