Re: The set of necessary FISONs

On 2/25/25 4:49 AM, WM wrote:

On 24.02.2025 19:25, Jim Burns wrote:

On 2/24/2025 12:10 PM, WM wrote:

 

In the present framework,
you (WM) confuse a claim about each FISON in {F}
with a claim about {F}.

 I never talked about {F}.

But {F} is a set of sufficient FISON whose union make ℕ, so you should be.
The fact the set of REQUIRED FISONs is empty, which is what you go off to try to prove is irrelevant, as having values required to get to the results only happen limited cases.
There are no REQUIRED factors of 36, as we can factor it many ways.
The prime number 23 is different, you can't make an Natural Number factoring of it without using the number 23.
So, all you have proven is that there isn't a highly restiricted path to building up the set of Natural Numbers from FISONs, any infinite set of them will do, and since there isn't a "highest" that needs to be included to get the "last" value, because infinite sets don't have such things, none become required.
Your logic that says all sets of numbers have a highest element is just broken, and its illogic has blown up your mind so you can't think rationally.

>

Zermelo creates all natural numbers by induction
and by that guarantees the existence of the set ℕ.

>
I guarantee that Zermelo was a finite being
and that, as such, he did not perform any supertask.

 Therefore he used induction.

But what you are calling "induction" isn't what is actually called induction

>
The existence of the set which
is its.own.only.inductive.subset
is proven from Zermelo's axioms.
We call that set ℕ.

 Defined by induction.

>

{1,2}\{1,{2}} = {2}

>
Only such nonsense available?

>
I'll grant you that it's trivial.
You (WM) have made it necessary to cover this.

 No.Your massive misunderstanding shows up above. I never used {ℕ} or {F}.

But {ℕ} is not ℕ. ℕ is the set { 0, 1, 2, 3, ... }
You keep on confusing sets for their elements, because you just don't understand them.

 Regards, WM