Re: The set of necessary FISONs

On 1/28/2025 4:13 AM, WM wrote:

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, ...} = { }

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What is ⋃{F(n)} ?

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It is the union of FISONs,  ℕ_def.

We all know what the union of FISONs is.
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
We all know what a FISON is.
A FISON has a natural order such that,
for each split,
its foresplit is empty or holds a foresplit.max i,
its hindsplit is empty or holds a hindsplit.min j,
i+1 = j
the FISON.minimum is first.natural,  and
each fuller.by.one set is larger and
each emptier.by.one set is smaller.
For each set such that
each fuller.by.one set is larger and
each emptier.by.one set is smaller,
a larger FISON exists.

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

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ℕ

We seem to disagree about what ℕ is.

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

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Yes, it is the dark domain.

𝔻ₘᵢₙ is the emptiest appendix such that
𝔻ₘᵢₙ completesᵂᴹ ⋃{F(n)} to {1,2,3,...}
which means
⎛ ⋃{F(n),𝔻ₘᵢₙ} \ ⋃{F(n),𝔻ₘᵢₙ}  =  {}
⎜ ∀n ∈ ⋃{F(n)}:
⎝ | ⋃{F(n),𝔻ₘᵢₙ} \{1,2,3,...,n}| = |⋃{F(n),𝔻ₘᵢₙ}|
A fuller.than.emptiest completingᵂᴹ.appendix 𝔻′
does not completeᵂᴹ what's already completeᵂᴹ.

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

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But potentially infinite.

The sum {1,...,j,j+1,...,j+k} of
two FISONs {1,...,j}, {1,...,k}
is a FISON.
⎛ For each set such that
⎜ each fuller.by.one set is larger and
⎜ each emptier.by.one set is smaller,
⎝ a larger FISON exists.
For each end.segment ⋃{F(n)}\{1,...,j}
a larger FISON does NOT exist,
|{1,...,k}| = |{j+1,...,j+k}|
For each end.segment ⋃{F(n)}\{1,...,j}
each fuller.by.one set is NOT larger and
each emptier.by.one set is NOT smaller.

No 𝔻 completesᵂᴹ ⋃{F(n)}

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Then actual infinity does not exist.

A set such that a larger FISON does NOT exist
is sufficiently large in order for Bob
to disappear purely from swaps within the set.
Any superset of such a set is also
a set such that a larger FISON does NOT exist
and is also
sufficiently large in order for Bob
to disappear purely from swaps within the set.
I agree that
your (WM's) actualᵂᴹ infinity does not exist.
That's not our infinity.