Re: The set of necessary FISONs

On 3/4/2025 2:56 PM, WM wrote:

On 04.03.2025 17:43, Jim Burns wrote:

On 3/4/2025 5:49 AM, WM wrote:

Here is only *one* argument standing for a long while.

>
The union ⋃{F} of FISONs and
the intersection ⋂𝒫ⁱⁿᵈ of inductive subsets
are the same set.

>
Correct.

Proper supersets of ⋂𝒫ⁱⁿᵈ contain extra elements.
You (WM) have previously said that
your (WM's) ℕ doesn't have extra elements.

You need not the intersection however because
Z₀ can also be defined by
{ } ∈ Z₀, and
if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀.

No.
For Z, not Z₀, ∀a: a ∈ Z ⇒ {a} ∈ Z
Z₀ is the simplest inductive subset of Z.
But you say "n+1 curly brackets".
I think you are trying to sneak ℕ in
through the back door.
If you want to say "finitely.many"
then say "finitely.many".
But say or accept what 'finitely.many' means.
Plus, "finitely.many" eliminates your darkᵂᴹ numbers.

Your (WM's) darkᵂᴹ numbers
are bracketed by ⋂𝒫ⁱⁿᵈ and ⋃{F}

>
No, they are following.

ω/2 ω/10 and ω/20 follow ω?

Every induction surpasses every finite number but never all finite numbers.
It is always a finite number.
Cantor's ℕ however is larger:
"Unter einem A.-U. ist dagegen ein Quantum zu verstehen,
das einerseits nicht veränderlich,
sondern vielmehr in allen seinen Teilen fest und bestimmt,
eine richtige Konstante ist,
zugleich aber andrerseits jede endliche Größe
derselben Art an Größe übertrifft. Als Beispiel führe ich die Gesamtheit,
den Inbegriff aller endlichen ganzen positiven Zahlen an;
diese Menge ist ein Ding für sich und bildet,
ganz abgesehen von der natürlichen Folge
der dazu gehörigen Zahlen,
ein in allen Teilen festes, bestimmtes Quantum,
... das offenbar größer zu nennen ist als jede endliche Anzahl.

⎛ "By an A.-U., on the other hand, we understand
⎜ a quantum that is on the one hand not variable,
⎜ but rather fixed and determined in all its parts,
⎜ a true constant,
⎜ but at the same time
⎜ on the other hand exceeds in size
⎜ any finite quantity of the same kind.
⎜ As an example, I cite
⎜ the totality, the sum of all finite whole positive numbers;
⎜ this set is a thing in itself and, quite apart from
⎜ the natural sequence of the numbers belonging to it,
⎜ forms a quantum that is fixed and determined in all its parts,
⎝ ... which can obviously be called larger than any finite number.
-- google

Therefore ℕ is more than UF.
By induction like Zermelo we find:
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo.

∀n ∈ ⋃{F}: |⋃{F}\{1,2,3,...,n}| = ℵ₀
For each and only finite sets A, there is
a terminating initial segment [0,k] of ⋃{F}
which is larger than A.
For each and only finite sets A,
a fuller.by.one set Aᣕᵃ is larger
A set is not larger than its superset.
⎛ Assume ⋃{F} is finite.
⎜
⎜ There is terminating initial segment [0,k]
⎜ larger than ⋃{F} and also,
⎜ because [0,k] ⊆ ⋃{F},
⎜ not larger than ⋃{F}
⎝ Contradiction.
⎜ but at the same time
⎜ on the other hand exceeds in size
⎜ any finite quantity of the same kind.
For example, like ⋃{F}.
⋃{F} is infinite.

⎛ ⋂𝒫ⁱⁿᵈ = ⋃{F}

>
True.

ω/2, ω/10, ω/20 ∈ ⋂𝒫ⁱⁿᵈ\⋃{F} = {}

⎜ Mehdi Hasan, a British journalist, suggests using
⎜ three steps to beat the Gish gallop:

https://en.wikipedia.org/wiki/Gish_gallop