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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 12 Feb 2025 10:19:54 +0100
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On 12.02.2025 01:38, Jim Burns wrote:
> On 2/11/2025 2:23 PM, WM wrote:
>> On 11.02.2025 18:42, Jim Burns wrote:
>>> On 2/11/2025 4:31 AM, WM 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.
> 
> What is this "both proofs"?

One is Zermelo's proof by induction that there is an infiite set Z.
The other is my proof by induction:
Assume that the set F of FISONs F(n) = {1, 2, 3, ..., n} has the union
UF = ℕ.
Notice that F(1) can be omitted without changing the result.
Notice that when F(k) can be omitted, then also F(k+1) can be omitted.
This makes the set of FISONs which can be omitted without changing the 
result an inductive set. It has no last element. It is F. The 
complementary set of FISONs which cannot be omitted, has no first 
element. It is empty.
 From the assumption UF = ℕ we have obtained U{ } = { } = ℕ. This result 
is false. By contraposition we obtain UF ≠ ℕ.

>> There is no further step.
> 
> Then there is no further conclusion.

 From the assumption UF = ℕ we have obtained U{ } = { } = ℕ.
That is no step but the result of the proof by induction.

> Be so kind as to cease claiming U{} = N

Try to improe your mathematical abilities. If the complete set can be 
omitted, then nothig remains. For contraposition consult any textbook, 
for instance W. Mückenheim: "Mathematik für die ersten Semester", 4th 
ed., De Gruyter, Berlin (2015)

>> Claimed is only UF = ℕ ==> ⋃{} = ℕ.
> 
> What is your new reason for claiming UF = U{} ?

It is not new but proved by induction.
> 
> ⎛ Not that Unicode character U+2115 'ℕ'
> ⎝ is as good as any other name for UF.

Please learn about induction. For instance from my book W. Mückenheim: 
"Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin 
(2015). But the first three editions are also available.

>> What is wrong in my application in your opinion?
> 
> You (WM) currently are ignoring 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′
> which means
> you ignore that
> for each FISON F'
> the union of FISONs.after F' are equal.
> which contradicts U{F} = U{}

Proofs by induction have no reason to observe your claim. They speak for 
themselves. Learn induction. Again I recommend my book. There is no 
reason to consider your claims. Further also Zermelo did not. Therefore 
you have lied.
> 
> More broadly,
> you (WM) write
>> Induction covers the whole infinite set.
> You mean by that
> that P(k) such that P(0) ∧ ∀k:P(k)⇒P(k+1)
> is valid for the minimal.inductive set itself
> in addition to each of its elements.
> 
> Which is usually not.even.wrong.

If Zermelo's induction is valid for his set Z including the set ℕ, then 
my proof is  valid for the set F too. If both proofs are valid for all 
elements only, then the result is also sufficient.

Regards, WM