Re: The set of necessary FISONs

On 2/24/2025 9:59 AM, WM wrote:

On 23.02.2025 23:03, Jim Burns wrote:

On 2/23/2025 2:32 PM, WM wrote:

On 23.02.2025 19:34, Jim Burns wrote:

On 2/23/2025 9:43 AM, WM wrote:

There is no reason to consider {{F}} at all.

>
There is reason, but
only for people wantcing to be correct.

>
Peano, Zermelo, or v. Neumann

>
...agree that {{F}} ≠ {F}

>
and also that 3 ≠ pi.

<WM<JB>>

>
{{}} ≠ {}

>
Wrong.
>

</WM<JB>>
Peano, Zermelo, v. Neumann: {{}} ≠ {}
⎛ Date: Sun, 23 Feb 2025 15:43:48 +0100
⎝ Message-ID: <vpfc74$hq5c$2@dont-email.me>

Peano, Zermelo, or v. Neumann create ℕ

>
Peano, Zermelo, and v. Neumann assert axioms
from which the existence of ℕ follows
in a finite.sequence of not.first.false claims.

>
These axioms can be applied to show that
all FISONs can be removed.

{1,2}\{1,{2}} = {2}
{2) ∉ {1,2)
2 ∈ {1,2)
∀S ∈ {F}: ⋃{F:S⊆F} = ⋃{F}
∀S ∈ {{F}}: ¬(⋃{F:S⊆F} = ⋃{F})
∀S ∈ {F}: {F:S⊆F} ≠ {}
∀S ∈ {{F}}: S = {F} ∉ {F}

as well as the set F of all FISONs
by induction over the members
for use in set theory
without being what you erroneously call correct.

>
A proof.by.induction shows that
some set,
such as the set {x:A(x)} of x such that A(x),
is inductive.
>
The conclusion of a proof.by.induction
is that {x:A(x)} is the whole set.
>
However,
not just any "whole set" is reliable here.
It must be a whole set such that
knowing {x:A(x)} is inductive
narrows
which set {x:A(x)} can be
to one set: that whole set.

>
The set of finite ordinals after v. Neuman
is undoubtedly such a set.

The set of finite ordinals after v. Neumann
is not a finite set.
A claim for each of its elements is
silent about the set.

We omit all F(n) which amounts to remove F.
Like all natural numbers amount to ℕ (not {ℕ})

>
Each natural number is in the domain of ST+F
ℕ is not in the domain of ST+F

>
Only all FISONs = natural numbers are
the matter of my proof.
According to Zermelo they make up the set ℕ.

If an object is in ℕ, it is a natural number.
If an object is a natural number, it is in ℕ.
If an object is in ℕ and ≥ 2,
it has a unique prime factorization.
If an object ends a FISON
⎛ for each FISON.split,
⎜ its foresplit ends at i or is empty
⎜ its hindsplit starts at j or is empty,
⎜ i+1 = j
⎝ The FISON starts at 0 and ends at the object,
it is in ℕ.
ℕ does not have a unique prime factorization.
ℕ does not end a FISON
Claims about each natural number
are silent about ℕ.