Re: how

On 6/22/2024 5:30 PM, WM wrote:

Le 22/06/2024 à 23:04, Jim Burns a écrit :

U{FISON} is the union of all its predecessors.
U{FISON} is not a FISON. It is not finite.

>
The union cannot be larger than all its FISONs.

The union cannot be smaller than any FISON.
¬∃F[k]:  ⋃{FISON} ̊< F[k]
∀F[k]:  F[k] ̊≤ ⋃{FISON}
Each FISON is smaller than another FISON,
another which the union cannot be smaller than,
each which the union is larger than.
∀F[j] ∃F[k]:  F[j] ̊< F[k]
∀F[j] ∃F[k]:  F[j] ̊< F[k] ̊≤ ⋃{FISON}
The union is larger than each FISON.
∀F[j]:  F[j] ̊< ⋃{FISON}

Yes,
U{FISON} is the union of all its predecessors.
No,
U{FISON} is not a FISON.

>
How can adding a FISON
change the character of the union?

Adding one FISON doesn't change the character.
Adding one isn't adding all FISONs.
Each FISON is followed by ℵ₀.many FISONs.
Adding all ℵ₀.many followers changes the character.
----
Each FISON is followed by ℵ₀.many FISONs.
S(F[j]) = F[j⁺¹]  is 1.to.1
S:{F[j]…} ⇉ {F[j⁺¹]…}: 1.to.1
id:{F[j⁺¹]…} ⇉ {F[j]…}: 1.to.1
{F[j]…} ̊≤ {F[j⁺¹]…} ̊≤ {F[j]…}
{F[j]…} ̊= {F[j⁺¹]…}
There is NO FIRST ̊<.ℵ₀.follower.FISONs FISON
¬∃#1F[j⁺¹]:
{F[j⁺¹]…} ̊< ℵ₀
{F[j]…} ̊= ℵ₀
{F[j]…} ̊≠ {F[j⁺¹]…}
There is NO ̊<.ℵ₀.follower.FISONs FISON
¬∃F[j+1]:  {F[j+1]..} ̊< ℵ₀
Therefore,
each FISON is followed by ℵ₀.many FISONs.

Every existing [nonempty] set of ordinals has
a first or smallest element.
The set of FISONs necessary to yield ℕ would be
[an empty] set of ordinals.
It is well defined since
for every FISON we can determine
whether it is necessary. It has no smallest element.
==> It is empty.

Yes,
the set of FISONs necessary to yield ℕ is empty.
U{FISON}  is the union of the set of
both necessary and unnecessary FISONs
FISON.union U{FISON} and minimal.inductive ⋂{inductive}
are both inductive and both well.ordered.
U{FISON} = ⋂{inductive}