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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 5 Mar 2025 10:22:06 +0100
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In-Reply-To: <4ae7b6d4-49a9-47ba-b2ac-c77238e93545@att.net>

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