Re: The set of necessary FISONs

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 wanting to be correct.

>
Peano, Zermelo, or v. Neumann

 ...agree that {{F}} ≠ {F}

and also that 3 ≠ pi.

 

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.

>

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.

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.
Regards, WM