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

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Then number theory can be wasted, because its formula for n/logn primes in (0, n] is no longer valid after Cantor-reordering.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