Re: The set of necessary FISONs

On 2/11/2025 4:31 AM, WM wrote:

On 10.02.2025 16:16, Jim Burns wrote:

On 2/10/2025 4:56 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.
For any FISON F′
⎛ each FISON.number is before a FISON.end which
⎜ is after the FISON.end of F'
⎝ each FISON.number is in a FISON after F′
For any FISON F′
⎛ each FISON.number is in a FISON after F′
⎝ each FISON.number is in the union of FISONs after F′
For any FISON F′
⎛ F′ is a FISON omissible
⎝ without changing the FISON.union.

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.
Only a hypothetical last.FISON ͚F,
a FISON with {after.͚F} = {}
supports your reasoning:
ᵂᴹ⎛ If, in general, ⋃{after.F′} = ℕ
ᵂᴹ⎜ and {after.͚F} = {}
ᵂᴹ⎝ then ⋃{after,͚F} = ⋃{} = ℕ
There is no last FISON.
⋃{} ≠ ℕ
Set theory lives another day.

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.

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′
?

How induction works is well known.

The passive voice leaves who it is who knows
unspoken. It matters who.
How induction works is well known to many people.
Yes, to that.
How induction works is not well known to you (WM).
No, to that.
⎛ I am still of the opinion that
⎜ you (WM) could know how induction works
⎜ (subjunctive mood),
⎜ but it possible, maybe even likely,
⎜ that you (WM) will go to your grave not knowing.
⎜
⎜ Induction is not that difficult, technically, but
⎜ your realization of a lifetime mis.spent
⎜ feeding bullshit to your students might well be
⎝ too steep a hill for you to ever climb.

If not consult Wikipedia or my book
W. Mückenheim: "Mathematik für die ersten Semester",
4th ed., De Gruyter, Berlin (2015)

Is that the textbook in which you teach that
ᵂᴹ( for some x, we can't say x = x
?