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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 11 Feb 2025 10:31:48 +0100
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On 10.02.2025 16:16, Jim Burns wrote:
> On 2/10/2025 4:56 AM, WM wrote:

>> The set of useless FISONs is inductive
>> and therefore infinite.
>> No FISON can change the assumption
>> U(A(n)) = ℕ.
>> Therefore every FISON can be omitted.
> 
> Do you accept
> ∀ᴺj′:∀ᴺi′:∃ᴺk′:
>   k′ = max{i′,j′+1}
> ?

No, I won't try to dive into your private notation. How induction works 
is well known. If not consult Wikipedia or my book
W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De 
Gruyter, Berlin (2015)

The set F of FISONs which can be removed without changing the assumed 
result UF = ℕ is the infinite set F of all FISONs. This is proven by 
just the same induction as Zermelo proves his infinite set Z.

Either you accept both proofs or none. But without there is no set theory.

Regards, WM