Re: The set of necessary FISONs

On 2/21/2025 1:04 PM, WM wrote:

On 21.02.2025 17:14, Jim Burns wrote:

On 2/21/2025 4:19 AM, WM wrote:

On 20.02.2025 20:46, Jim Burns wrote:

For the sets of all (finite) FISONs and
of all finite von Neumann ordinals,
(**) is satisfied as a consequence of
the finitude of their elements.

>
S = ℕ_def

>
ℕ_def = ℕ′
>
S = {i:A(i)}
>
⎛ A(1)  ∧  ∀n∈ℕ′:A(n)⇒A(n+1)   ⇒
⎜ {i:A(i)} ∈ {S″⊆ℕ′:inductive.S″}
⎜
⎜ {S″⊆ℕ′:inductive.S″} = {ℕ′}
⎜ ⇐ all and only finites are in ℕ′
⎜
⎜ ( {ℕ′} ≠ ℕ′ )

>
We need not consider {ℕ′}.

We consider {S″⊆ℕ′:inductive.S″} = {ℕ′}
and we find that it follows that
A(1) ∧ ∀k∈ℕ′:A(k)⇒A(k+1)  ⇒  ∀n∈ℕ′: A(n)

⎜ {i:A(i)} ∈ {S″⊆ℕ′:inductive.S″}  ∧
⎜ {S″⊆ℕ′:inductive.S″} = {ℕ′}  ⇒
⎜ {i:A(i)} = ℕ′
⎜
⎜ {i:A(i)} = ℕ′  ∧
⎜ ∀k ∈ {i:A(i)}: A(k)  ⇒
⎝ ∀k ∈ ℕ′: A(k)
>
Proof by induction for ℕ′
  ⇐ all and only finites are in ℕ′

>
And all together are the set ℕ′.

We consider a set which
is its own only.inductive.subset.
It should be very easy
to give that set
the same name everyone else gives it.