Re: Incompleteness of Cantor's enumeration of the rational numbers

Am Sat, 23 Nov 2024 23:01:00 +0100 schrieb WM:

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.

No. "Almost all" means everything except for a finite number.
If something remains you only shifted a finite number.

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in (0, n] is no longer valid after Cantor-reordering.

It is still valid for every finite interval.

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.

Bob is in none of them.

--
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.