Re: The set of necessary FISONs

On 2/13/2025 6:59 AM, WM wrote:

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.

Did you read it?
Did it say this?
⎛
⎜ Inductiveᶻ Z exists:
⎜ Powerset 𝒫(Z) = {S ⊆ Z} of Z exists
⎜ Set 𝒫ⁱⁿᵈ(Z) of inductiveᶻ subsets of Z exists
⎝ Intersection ⋂𝒫ⁱⁿᵈ(Z) of inductiveᶻ subsets of Z exists
⎛ Define ℕ = ⋂𝒫ⁱⁿᵈ(Z)
⎜ ∀P ∈ {T,F}ᴺ:
⎝ P(0) ∧ ∀ᴺk:P(k)⇒P(k+1) ⇒ ∀ᴺn:P(n)
Those proofs support later proofs by induction.
They aren't proofs by induction.

But that is irrelevant for the present topic.

The present topic is "What is induction?"

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

each FISON

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.

ℕ is minimal.inductive, meaning
ℕ is the only inductive subset of ℕ
Prove that P is an inductive subset of ℕ
and prove that P is ℕ

Mathematical induction is
a method for proving that
a statement is true for every natural number

You (WM) think
ᵂᴹ⎛ that ω-1 exists
ᵂᴹ⎜ that ℕ is a FISON
ᵂᴹ⎜ that the set of priors of
ᵂᴹ⎜  a number of the second Cantor class
ᵂᴹ⎝ has emptier.by.one sets smaller.
The present topics are "What is induction?"
and "What is a natural number?"
 visibleᵂᴹ darkᵂᴹ matheologyᵂᴹ
    ↕        ↕        ↕
  finite   finite  infinite

{P(0),P(1),P(2),P(3),\dots }  all hold.
>
No FISON remains.