Re: The set of necessary FISONs

On 1/21/2025 6:45 AM, WM wrote:

All finite initial segments of natural numbers,
FISONs F(n) = {1, 2, 3, ..., n}
as well as their union
are less than the set ℕ of natural numbers.

Natural numbers are finite ordinals.
The finites extend infinitely further
than you (WM) think they extend.
Only for a finite set,
fuller.by.one sets are larger  and
emptier.by.one sets are smaller.
For each finite set, there is
a (finite) FISON larger than it,
and that FISON subsets ℕ
For ℕ
there is no FISON larger than it
therefore,
ℕ isn't finite,
therefore,
for ℕ
fuller.by.one sets are NOT larger  and
emptier.by.one sets are NOT smaller.

Proof:
Assume UF(n) = ℕ.
The small FISONs are not necessary.
What is the first necessary FISON?
There is none!

For each finite set, there is
a (finite) FISON larger than it.
For each (finite) FISON, there is
a (finite) FISON larger than it,
and it can be dropped.

All can be dropped.

Each is not all.
Each can be dropped.
All cannot be dropped.

But according to Cantor's Theorem B,
every non-empty set of different numbers of
the first and the second number class
has a smallest number, a minimum.
This proves that
the set of indices n of necessary F(n),
by not having a first element,
is empty.

ℕ = ⋃{FISON}
∀F ∈ {FISON}:
∃F′ ∈ {FISON}:
F′ = F+1 := F∪{1+max.F}
F ≠⊂ F′
∀F ∈ {FISON}:
⋃{F+1,F} = ⋃({F+1,F}\{F})
⋃{FISON} = ⋃({FISON}\{F})
Each F is unnecessary to ⋃{FISON}
{unnecessary.FISON} = {FISON}
∀F ∈ {unnecessary.FISON}:
∃F′ ∈ {unnecessary.FISON}:
F′ = F+1 := F∪{1+max.F}
F ≠⊂ F′
∀F ∈ {unnecessary.FISON}:
⋃{unnecessary.FISON} = ⋃{{unnecessary.FISON}\{F})
Each F is unnecessary to ⋃{unnecessary.FISON}
{unnecessary.unnecessary.FISON} = {unnecessary.FISON}
----
For each unnecessary.FISON F
there is an unnecessary.FISON.swap F⇄F+1
from F to (unnecessary) F+1
They can be ordered such that
  F⇄F+1 precedes F₂⇄F₂+1  ⇔  F ≠⊂ F₂
For each unnecessary.FISON.swap F-1⇄F
into F, there is
a later unnecessary.FISON.swap F⇄F+1
out of F
Consider Bob such that,
before all FISON.swaps,
Bob is in the first FISON ℕ
If Bob is in FISON F
it is after F-1⇄F and before F⇄F+1
If it is after all unnecessary.FISON.swaps
then Bob is not.in any FISON,
even though
no unnecessary.FISON.swaps take Bob
anywhere else.
KING BOB!!!
https://youtu.be/jK2XzKDab0E?si=ZROiFjKMTmiaGqPz&t=43