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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 20:23:28 +0100
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On 11.02.2025 18:42, Jim Burns wrote:
> On 2/11/2025 4:31 AM, WM wrote:
>> On 10.02.2025 16:16, Jim Burns wrote:

>> The set F of FISONs which can be removed without
>> changing the assumed result UF = ℕ
>> is the infinite set F of all FISONs.
> 
> Yes.

Fine.
>> 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.
> 
> That part above is fine.
> Your problem is in the step after that,
> which, for some reason, you skip over.

The part above defines the set of all FISONs that can be omitted without 
changing the union. There is no further step.
> 
> Only a hypothetical last.FISON ͚F,
> a FISON with {after.͚F} = {}
> supports your reasoning:

The set of all FISONs is accepted. No further restricitons allowed.

> There is no last FISON.
> ⋃{} ≠ ℕ

That is not claimed. Claimed is only UF = ℕ ==> ⋃{} = ℕ.

Can't you understand the meaning of an implication?
>> No, I won't try to dive into your private notation.
> 
> Do you accept that,
> for each two FISON.numbers j′ and i′
> there exists a FISON.number maximum k′ of
> i′ and the successor j′+1 of j′
> ?
It is irrelevant what details exist. Induction covers the whole infinite 
set.

> How induction works is not well known to you (WM).

What is wrong in my application in your opinion?

Regards, WM