Re: The set of necessary FISONs

On 27.01.2025 23:07, Jim Burns wrote:

On 1/27/2025 7:55 AM, WM wrote:

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

What is ⋃{F(n)} ?

It is the union of FISONs,  ℕ_def.

What is {1,2,3,...} ?

ℕ

{1,2,3,...} = (⋃{F(n)})∪𝔻 = ⋃{F(n),𝔻}
 Certainly, an appendix 𝔻 is possible.

Yes, it is the dark domain.

∀n ∈ ⋃{F(n)}:
|⋃{F(n)}\{1,2,3,...,n}| = |⋃{F(n)}\{1,2,3,...,n-1}|
 There is no first end.segment smaller than ⋃{F(n)}
The end.segments are well.ordered.

But potentially infinite.

No 𝔻 completesᵂᴹ ⋃{F(n)}

Then actual infinity does not exist.
Regards, WM