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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 3 Mar 2025 17:38:21 +0100
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On 03.03.2025 12:54, joes wrote:
> Am Mon, 03 Mar 2025 09:57:47 +0100 schrieb WM:

>> Z₀ does not contain undefinable elements. It contains simply all numbers
>> that have FISONs (like v. Neumann constructs the FISONs directly).
> No. Z_0 is equivalent to N.

How that? By induction we prove for every element n ∈ Z₀:
|ℕ \ {1, 2, 3, ..., n}| = ℵo
> sqrt(2) has an infinite decimal representation, like all reals. It just
> doesn't have the period (0).

There is no actually infinite decimal representation. It would have ω 
digits. What digit would be at position ω/2?
> 
>>> Therefore, two such decimal.splitting.points √2 and √2′ don't exist.
>> Yes, but the topic is this: In exactly the same way as Z₀ is constructed
>> by its elements, the set of removable FISONs is constructed by its
>> elements, namely by induction.
> And in the same way that Z_0 doesn't contain infinite elements, the set
> of removable sets of FISONs doesn't.

Of course not. Only finite FISONs are existing and removable. If all are 
removed, none remains. UF = ℕ  ==> Ø = ℕ.

Regards, WM
>