Re: The set of necessary FISONs

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