Re: The set of necessary FISONs

On 25.01.2025 20:35, Jim Burns wrote:

If it is after all swaps,
then Bob isn't in any FISON.end room,
even though swaps only place him in FISON.end rooms.

He does not disappear because lossless exchanges do not allow losses. Hence he must be in the dark domain.

 

Do you agree that
for every FISON
the question whether it is necessary can be answered?

 If Bob is somewhere after all swaps,
how did he get there?

By lossless exchanges.

 

Swaps cannot eliminate Bob.
He remains but i the darkness.

 No swaps are into the darkness.
 

No swaps executed consciously can end in darkness. But swaps produced by a general formula must end in darkness because ℵo numbers lurk behind all definable numbers: ∀n ∈ U(F(n)): |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Regards, WM