Re: The set of necessary FISONs

On 2/6/2025 2:32 PM, WM wrote:

On 06.02.2025 19:54, Jim Burns wrote:

On 2/6/2025 11:55 AM, WM wrote:

On 06.02.2025 15:57, Jim Burns wrote:

The key is that  ∀ᴺ¹n: ∃ᴺ¹j′: n<j′

>
The key is that
the set ℕ is created by induction.

>
The set ℕ₁ is described as having induction valid for it.

>
Then it is the collection ℕ_def of definable numbers.
>

Sets missing natural numbers and
sets with extra, non.inducible, un.natural numbers
are not ℕ₁

>
Then it is the set ℕ of all natural numbers.
>
You contradict yourself.

No.
One set is both
the emptiest superset ⋃{F} of each FISON and
the fullest set ⋂𝒫ⁱⁿᵈ common to all inductive sets.
----
⎛ The fullest set ⋂ℐ common to a set ℐ of inductive sets
⎜ is inductive.
⎜
⎜ For each inductive set 𝕂, there is
⎜ an inductive set ⋂𝒫ⁱⁿᵈ(𝕂) common to
⎜ all that set's inductive subsets 𝒫ⁱⁿᵈ(𝕂)
⎜
⎜ For each two inductive sets 𝕂 and 𝕄
⎜ ⋂𝒫ⁱⁿᵈ(𝕂) = ⋂𝒫ⁱⁿᵈ(𝕄) = ⋂𝒫ⁱⁿᵈ
⎜
⎜ (Inductive) ⋂𝒫ⁱⁿᵈ is
⎜ the fullest set common to all inductive sets.
⎜
⎝ If 𝕏 is inductive, then ⋂𝒫ⁱⁿᵈ ⊆ 𝕏
⎛ 0 ∈ ⋃{F}
⎜ k ∈ ⋃{F} ⇒ k+1 ∈ ⋃{F}
⎜ ⋃{F} is inductive.
⎜
⎝ ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎛ No element of any FISON is not.in ⋂𝒫ⁱⁿᵈ
⎜
⎜ No element of ⋃{F} is not.in ⋂𝒫ⁱⁿᵈ
⎜
⎝ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎛ ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎜
⎜ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎜
⎝ ⋂𝒫ⁱⁿᵈ = ⋃{F}