Re: The set of necessary FISONs

On 26.01.2025 23:54, Jim Burns wrote:

On 1/26/2025 4:49 AM, WM wrote:

No swaps can complete actual infinity of ℕ.

 

No actual infinity is represented by the first column.

 No appendix 𝔻 such that ∀d ∈ 𝔻: f(d) = d
completes potential.infinity to actual.infinity.

∀n ∈ U(F(n)): |ℕ \ {1, 2, 3, ..., n}| = ℵo
ℕ \ {1, 2, 3, ...} = { }

No swaps can complete actual infinity of ℕ.

 For each FISON, there is
a fuller.by.one FISON which is larger.
That is complete enough
in order to allow Bob to disappear.

Not outside of all numbers.
All the FISONs are smaller than ℵo elements:
{1}
{2, 1}
{3, 2, 1}
...
which prove that the first column is smaller than ℵo elements too.
That means no actual infinity is involved in visible numbers. Bob may go further. Then he becomes invisible.

The sets aren't larger and aren't smaller.
They are the same size,
which allows Bob to disappear by lossless swaps.

Yes, in the dark domain.
Regards, WM