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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 12 Mar 2025 21:09:08 +0100
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On 12.03.2025 19:18, Jim Burns wrote:
> On 3/12/2025 9:04 AM, WM wrote:

>>> ⎛ Menge Z, welche die Nullmenge als Element enthält und
>>> ⎜ so beschaffen ist, daß jedem ihrer Elemente a
>>> ⎝ ein weiteres Element der Form {a} entspricht
> 
> ⎛ set Z, which contains the zero set as an element and
> ⎜ is such that each of its elements a
> ⎝ corresponds to another element of the form {a}
> 
>> That is just the induction.
> 
> A definition answers "What is Z ?"
> An axiom answers "Does Z exist?"

The axiom ensure the existence of an infinite set by induction.
Only that is of interest for comparison with my proof.
> The axiom does not create an inductive set.

It ensures its existence.
> Z₀ -- defined as the emptiest inductive set --
> doesn't hold darkᵂᴹ {}  and
> doesn't hold any visibleᵂᴹ x and darkᵂᴹ {x}

Of course not. Inductive sets do not contain dark numbers.

> 
> {darkᵂᴹ} = Z₀\{visibleᵂᴹ} = {}

Nonsense. Dark is Cantor's ℕ \ Z₀.
> 
>> my induction is ensured
> 
> Your induction is not ensured by
> your declaration that it's ensured.

It is ensured by using induction.

> How is your induction ensured?

Here you can see it:
> 
>>>> ℕ \ F(1) = ℵo,
>>>> and if
>>>> ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n) = ℵo
>>>> then
>>>> ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n+1) = ℵo.
> 
Regards, WM