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

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

On 24.10.2024 19:34, Jim Burns wrote:

On 10/24/2024 6:58 AM, WM wrote:

All that is accepted and doubled.

>
The subset of those with a double not.in ℕ
holds no minimum.

>
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.

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.

Note well:
After all _infinitely many_ swaps,
Bob is not in any of _infinitely many_ rooms
he was ever in.
>
Infinite is different from finite.

>
Bobs cannot dissolve into nothing.

Bob can be in an infinite set.

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.