Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

Am 25.10.2024 um 06:33 schrieb Jim Burns:

On 10/24/2024 2:58 PM, WM wrote:

The whole set with all its numbers exists
and can be mapped to the double numbers.

 Yes.
∃⟨0,1,...,n-1,n⟩  ⇒
∃⟨0,1,...,n-1,n,n+1,...,n+n-1,n+n⟩
 

That halves the density
and doubles the covered interval.

 No.
⟨0,1,...,n-1,n,n+1,...,n+n-1,n+n⟩ is finite.

The whole interval (0, ω) is not finite, let alone the doubled interval.

 

After all swaps,
Bob is not in any room he was ever in.
'Bye, Bob.

>
After all swaps Bob is in a room,
because there is nowhere an outlet.

 That Bob is not in any room he was ever in
and there is nowhere an outlet
proves the set is infinite.

Bob of the matrices
XOOO... XXOO... XXOO... XXXO... ... XXXX...
XOOO... OOOO... XOOO... XOOO... ... XXXX...
XOOO... XOOO... OOOO... OOOO... ... XXXX...
XOOO... XOOO... XOOO... OOOO... ... XXXX...
...........................................
has come to rest if all natural numbers have been applied.

 

Bobs cannot dissolve into nothing.

 Bob can be in an infinite set.

If the infinite set has been used completely for counting then Bob stays in a dark cell.

 

The problem with Bob disappearing is that
Bob disappearing is not a problem.
It's not repairable because it's not broken.

>
Lossless exchanges remain lossless forever.

 After all swaps,
Bob is not in any room he was ever in.

If all natnumbers can be issued, Bob is in a dark cell because further nothing happens.
If you dislike this result, then you oppose to complete infinity.
Regards, WM
Regards, WM