Re: The set of necessary FISONs

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"?
You say there is no further step.
You:

There is no further step.

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.

Then there is no further conclusion.
Be so kind as to cease claiming U{} = N

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.

Then
be so kind as to cease making further claims.

There is no last FISON.
⋃{} ≠ ℕ

>
That is not claimed.

No.
You:
⎛ Therefore U(A(n)) = ℕ ==> U{ } = { } = ℕ.
⎝
Message-ID: <vo8jr7$7mh2$2@dont-email.me>
Date: Sat, 8 Feb 2025 22:54:46 +0100

Claimed is only UF = ℕ ==> ⋃{} = ℕ.

What is your new reason for claiming UF = U{} ?
⎛ Not that Unicode character U+2115 'ℕ'
⎝ is as good as any other name for UF.

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?

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{}
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.
I'll save the rest for another post, sometime.