Re: The set of necessary FISONs

On 12.02.2025 20:39, Jim Burns wrote:

On 2/12/2025 4:19 AM, WM wrote:

One is Zermelo's proof by induction
that there is an infiite set Z.

 https://en.wikipedia.org/wiki/Zermelo_set_theory
 The proof of Z you refer to is the bare assertion
that inductiveᶻ Z exists, AXIOM VII "Infinity".
  Z exists such that Z∋{} ∧ ∀a∈Z∋{a}

Then follows almost a whole page of proof. But that is irrelevant for the present topic.

From the assumption UF = ℕ
we have obtained U{ } = { } = ℕ.

We have obtained that,
for each FISON F′ ∈ {F} and 𝒜 ∈ 𝒫{F}
if ⋃𝒜 = ℕ then _not.necessarily_ F′ ∈ 𝒜

All of us who know induction know that by induction we have obtained that all FISONs can be removed without changing the result.

From the assumption UF = ℕ
we have obtained U{ } = { } = ℕ.

  From the assumption ⋃{F} = ℕ
we have obtained ⋂𝒫ᵁᐧᙿᴺ{F} = {}
and also 𝒫ᵁᐧᙿᴺ{F} ≠ {}

Learn induction. Mathematical induction is a method for proving that a statement is true for every natural number {P(0),P(1),P(2),P(3),\dots }  all hold.
No FISON remains.
Regards, WM