Re: The set of necessary FISONs

On 04.03.2025 23:05, Jim Burns wrote:

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

We talk about subsets!

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

ℕ is a proper superset of ⋃F. It contains all the dark natural numbers. ⋃F contains only defined natural numbers.

 

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.

Ha. Caught red handed. Yes there is no mathematical possibility to contradict my claim.

 For Z, not Z₀, ∀a: a ∈ Z ⇒ {a} ∈ Z
Z₀ is the simplest inductive subset of Z.

What does Z₀ contain that is not in my inductive description?

 But you say "n+1 curly brackets".

You can also say: 0 and if n ∈ ℕ then n+1 ∈ ℕ. That is only another notation of number n.

Plus, "finitely.many" eliminates your darkᵂᴹ numbers.

a then {a} means precisely as many as n then n+1.>

ω/2 ω/10 and ω/20 follow ω?

No they follow Z₀ and UF.

 

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. (that abbreviates actual infinity), 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

Very good.

 

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

No. ⋃{F} is wrong because F is the set of FISONs. ⋃{F} = F.
UF does not contain an n that is larger than all n by ℵ₀.

⎛ Assume ⋃{F} is finite.

UF is (potentially in-) finite. It is constructed by induction which is variable and therefore has no largest number but never reaches a completion.
Regards, WM