Re: The set of necessary FISONs

On 03.02.2025 20:48, Jim Burns wrote:

On 2/3/2025 1:36 PM, WM wrote:

Therefore Peano, Zermelo, or v. Neumann
create ℕ as well as the set of all FISONs
for use in set theory.

 Axioms describe.
Magic spells create.

To describe something it must be existing. If ℕ is existing, we do not need axioms.

Therefore all FISONs can be removed from
the set of all FISONs.

 We can describe the removal of all of them, sic: {}

We can do it by induction: Remove F(1) and if you have removed F(n), remove F(n+1).

 

All natural numbers can be added by induction to a set A.

 Either all natural numbers are in A,
or they aren't all in A.
Those are all the choices.

Nonsense. Sets can be added and subtracted.  We can add the set ℕ to the set { } by adding 1, and if n has been added, then n+1 is added.

1 is added to A, and
if n is added to A, then n+1 is added to A.

 

All FISONs can be subtracted from the set of all FISONs
by the same procedure.
F(1) is subtracted.
If F(n) is subtracted, then F(n+1) is subtracted.

 For each FISON,
there is a larger FISON not larger than U{FISON}

When the set is subtracted by induction, nothing remains.

We could decide that  that isn't their behavior,
but, if we decide that, everything turns to gibberish.

Peano constructs by induction the set ℕ: If we add the set that contains 1 and with n also n+1, to { }, then we get ℕ.
If we subtract from ℕ the set that contains 1 and with n also n+1, then only { } remains. Same holds for FISONs. Therefore: if U(F(n)) = ℕ, then { } = ℕ.
Regards, WM