Re: The set of necessary FISONs

On 03.02.2025 19:06, Jim Burns wrote:

On 2/3/2025 7:41 AM, WM wrote:

How can Peano create the complete set by induction?

 Peano describes a set with induction.

Without axioms nothing must be used oin formal mathematics.
Therefore Peano, Zermelo, or v. Neumann create ℕ as well as the set of all FISONs for use in set theory.

It is a complete set which is described.
(We don't use any other, "incomplete" sets.)

Therefore all FISONs can be removed from the set of all FISONs.
All natural numbers can be added by induction to a set A. 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.

But the claims are silent about what wasn't described.
Peano describes _the elements_ of ⋃{FISON}
⋃{FISON} isn't an element of ⋃{FISON}

Peano creates the set of all natural numbers as well as the set of all FISONs.
Regards, WM