Re: The set of necessary FISONs

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

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, ...} = { }

You (WM) seem to be offering that as
a counter.example to my claim that there's no 𝔻
You (WM) seem to be using that
to say
ᵂᴹ⎛ U(F(n)) is _incompleteᵂᴹ_ in covering ℕ
ᵂᴹ⎜( ∀n ∈ U(F(n)): |ℕ\{1,2,3,...,n}| = ℵ₀
ᵂᴹ⎜ and {1,2,3,...} is _completeᵂᴹ_ in covering ℕ
ᵂᴹ⎜( ℕ\{1,2,3,...} = {}
ᵂᴹ⎜ because
ᵂᴹ⎜ there is an appendix 𝔻 = {1,2,3,...}\U(F(n))
ᵂᴹ⎜ which _completesᵂᴹ_ U(F(n)) to {1,2,3,...}
ᵂᴹ⎝ and that 𝔻 is offered as a counter.example.
What is ⋃{F(n)} ?
What is {1,2,3,...} ?
⋃{F(n)} is the union of all FISONs.
_What that means_ is that,
if 𝕏 is superset each FISON,
then ⋃{F(n)} is subset 𝕏 and superset each FISON
{F(n)} ᵉᵃᶜʰ⊆ 𝕏  ⇒  {F(n)} ᵉᵃᶜʰ⊆ ⋃{F(n)} ⊆ 𝕏
{1,2,3,...} = (⋃{F(n)})∪𝔻 = ⋃{F(n),𝔻}
Certainly, an appendix 𝔻 is possible.
However,
supposing your (WM's) counter.example to be correct,
only the _emptiest_ completingᵂᴹ 𝔻 completesᵂᴹ.
The fuller ones are fluff.
Since there is nothing left to accomplish,
the fuller ones accomplish nothing.
What is the _emptiest_ 𝔻ₘᵢₙ such that
⎛ ⋃{F(n),𝔻ₘᵢₙ} \ ⋃{F(n),𝔻ₘᵢₙ}  =  {}
⎜ ∀n ∈ ⋃{F(n)}:
⎝ | ⋃{F(n),𝔻ₘᵢₙ} \{1,2,3,...,n}| = |⋃{F(n),𝔻ₘᵢₙ}|
?
Emptiest 𝔻ₘᵢₙ = {}
because
⎛ ⋃{F(n),{}} \ ⋃{F(n),{}}  =  {}
⎜ ∀n ∈ ⋃{F(n)}:
⎝ |⋃{F(n),{}}\{1,2,3,...,n}| = |⋃{F(n),{}}|
For each set A such that
⎛ sets fuller.by.one than A are larger and
⎝ sets emptier.by.one than A are smaller
there is a FISON in {F(n)} the size of A
There is no FISON in {F(n)} the size of {F(n)}
There is no FISON in {F(n)} the size of ⋃{F(n)}
{F(n)} is NOT a set such that
⎛ sets fuller.by.one than {F(n)} are larger and
⎝ sets emptier.by.one than {F(n)} are smaller
⋃{F(n)} is NOT a set such that
⎛ sets fuller.by.one than ⋃{F(n)} are larger and
⎝ sets emptier.by.one than ⋃{F(n)} are smaller
A set the size of {F(n)} or ⋃{F(n)} is NOT a set such that
⎛ sets fuller.by.one than ⋃{F(n)} are larger and
⎝ sets emptier.by.one than ⋃{F(n)} are smaller
∀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.
There is no end.segment of ⋃{F(n)} smaller than ⋃{F(n)}
∀n ∈ ⋃{F(n)}:
|⋃{F(n)}\{1,2,3,...,n}| = |⋃{F(n)}|
And, of course,
⋃{F(n)}/⋃{F(n)}  =  {}
Pre.appendixed ⋃{F(n)} is as completeᵂᴹ and as incompleteᵂᴹ
as any ⋃{F(n),𝔻}
as ⋃{F(n),𝔻ₘᵢₙ}
as ⋃{F(n),{}}
as ⋃{F(n)}
No 𝔻 completesᵂᴹ ⋃{F(n)}

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

No appendix 𝔻 such that ∀d ∈ 𝔻: f(d) = d
completesᵂᴹ potentialᵂᴹ.infinity to actualᵂᴹ.infinity.