Re: The set of necessary FISONs

On 2/10/2025 4:56 AM, WM wrote:

On 09.02.2025 18:20, Jim Burns wrote:

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:

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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}
?

==> { } = ℕ.

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

The axiom of induction says:
If any property or predicate P satisfies
(P(1) /\ ∀k(P(k) ==> P(k+1)),
then it describes all elements of
an inductive = infinite set.

If any property or predicate P satisfies
(P(1) /\ ∀k(P(k) ==> P(k+1)),
then it describes all elements of
[a minimal.inductive] set.

minimal.inductive ≠ inductive ≠ infinite

>
Then you are wrong.
Every inductive set is infinite.

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Some infinite sets, such as E(137),
are not inductive.

>
That is irrelevant.

...to anyone not wanting to be correct.
Good to know what you find irrelevant.

Also,
some inductive sets, such as ℝ,
are not minimal.inductive.

>
That is irrelevant.

...to anyone not wanting to correctly.use

(P(1) /\ ∀k(P(k) ==> P(k+1)),

Good to know what you find irrelevant.