Re: The set of necessary FISONs

Am Mon, 24 Feb 2025 15:59:18 +0100 schrieb WM:

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.

Nope. Any number of them can be removed. Indeed, removing all
changes the union, as you have admitted.

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.

The set N is not a natural number that the proof concerns.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.