Re: The set of necessary FISONs

On 3/1/2025 1:28 PM, WM wrote:

On 01.03.2025 18:47, Jim Burns wrote:

On 3/1/2025 7:51 AM, WM wrote:

Z_0 contains only 0, {0}, {{0}}, ...

>
Z₀ = {0,{0},{{0}},...} is
the only subset of Z₀ which
holds 0 and, for each a, holds {a}

>
Z₀ is defined by induction.

inductive(Z)
inductive(Z) :⇔ 0∈Z ∧ ∀a:a∈Z⇒{a}∈Z
Z₀ is the emptiest (<EZ>"einfachste"?)
inductive set.
inductive(W) ⇒ Z₀ ⊆ W
inductive(Z₀)
Z₀ is not constructed by supertask.
Z₀ = ⋂𝒫ⁱⁿᵈ(Z)

Likewise UF is defined by induction.
ℕ \ F(1) \ F(2) \ F(3) \ ... = ℵo
= ℕ \ (F(1) U F(2) U F(3) U ...) = ℵo.
>

In narrower Z₀,
inductivity identifies a unique set.

>
So it is.

Thank you.
{S⊆Z₀:inductive.S} = {Z₀} ∧
inductive{i:A(i)}  ⇒
{i:A(i)} ∈ {SsZ₀:inductive.S} = {Z₀}  ⇒
Z₀ = {i:A(i)}
Z₀ = {i:A(i)} ∧
∀n ∈ {i:A(i)}: A(n)  ⇒
∀n ∈ Z₀: A(n)
Our inductive conclusions are justifiedly universal
because they are in narrower Z₀
not because of any supertask.