Re: The set of necessary FISONs

On 2/9/2025 5:59 AM, WM wrote:

On 09.02.2025 01:00, Jim Burns wrote:

On 2/8/2025 4:54 PM, WM wrote:

That is satisfied by the set M of
all FISONs which are useless in U(A(n)) = ℕ.

>
Without exception,
the union of FISONs.after is ℕ

>
No.

Do you accept
∀ᴺj′:∀ᴺi′:∃ᴺk′:  k′ = max{i′,j′+1}
?
⎛ ∀ᴺj′:
⎜ ∀i′ ∈ ℕ:
⎜ ∃ᴺk′:  k′ = max{i′,j′+1} ∧
⎜  i′≤k′ ∧ j′<k′ ∧
⎜  i′ ∈ {i:i≤k′} ∈ {{i:i≤k}:j′<k} ∧
⎜  i′ ∈ ⋃{{i:i≤k}:j′<k}
⎜
⎜ ∀ᴺj′:
⎜ ∀i′ ∈ ℕ:
⎜  i′ ∈ ⋃{{i:i≤k}:j′<k}
⎜
⎝ ∀ᴺj′:  ⋃{{i:i≤k}:j′<k} ⊇ ℕ
⎛ ∀ᴺj′:
⎜ ∀ᴺk′ > j′:
⎜ ∀ᴺi′ ≤ k′:  i′ ∈ ℕ
⎜
⎜ ∀ᴺj′:
⎜ ∀ᴺk′ > j′:
⎜ ∀i′ ∈ {i:i≤k′}:  i′ ∈ ℕ
⎜
⎜ ∀ᴺj′:
⎜ ∀i′ ∈ ⋃{{i:i≤k}:j′<k}:  i′ ∈ ℕ
⎜
⎝ ∀ᴺj′:  ⋃{{i:i≤k}:j′<k} ⊆ ℕ
⎛ ∀ᴺj′:  ⋃{{i:i≤k}:j′<k} ⊇ ℕ
⎜ ∀ᴺj′:  ⋃{{i:i≤k}:j′<k} ⊆ ℕ
⎝ ∀ᴺj′:  ⋃{{i:i≤k}:j′<k} = ℕ

Therefore U(F(n)) = ℕ ==>
U{ } = { } = ℕ.

>
Why that '==>' ?

>
From the assumption U(F(n)) = ℕ
I have derived that { } = ℕ.

That which you write after doesn't derive {} = ℕ.

My best guess at why you claim  U{ } = { } = ℕ
is that you (WM) are assuming that,
for some FISON (ie, F(ω-1)) such that
there are no FISONs.after.

>
No.
>

You (WM) haven't given any other reason.

>
If U(F(n)) = ℕ, then
F(1) can be omitted without changing the result.

The union of FISONs.after F(1) is ℕ

If F(k) can be omitted,
then F(k+1) can be omitted too.

If the union of FISONs.after F(k) is ℕ
then the union of FISONs.after F(k+1) is ℕ

The set of FISONs which can be omitted
is an inductive set, i.e., all FISONs.

For each FISON
the union of FISONs.after it is ℕ
That's true.
It's not the conclusion you (WM) want.
If one also assumes that there is
a FISON which ends the FISONs,
the set of FISONs.after it is {}
 From that assumption,
one gets the conclusion you (WM) want:
⋃{} = ℕ
However,
the FISONs are inductive.
No FISON ends the FISONs.
Perhaps you (WM) have decided that,
as long as you don't SAY you've assumed a claim
it doesn't count as an assumption.
Assuming doesn't work in that way.
Perhaps you (WM) have a different argument in mind.
I doubt you do, but, if you offer one,
I'll look at it.