Re: The set of necessary FISONs

On 02.02.2025 19:23, Jim Burns wrote:

F(1) can be discarded.
If F(n) can be discarded, then F(n+1) can be discarded.
>
Note:
Mathematical induction is a method for proving that
a statement P(n) is true for every natural number n
that is, that
the infinitely many cases P(0),P(1),P(2),P(3),...
all hold. [Wikipedia]

 For each k ∈ ⋃{FISON}:

No, for *all* k ∈ ⋃{FISON}.
Peano creates the set ℕ by induction.
I remove the set of FISONs by the same induction:
n ==> n+1.
What us the difference?

⋃{FISON} = ⋃({FISON}\{F(k)})
 P(k)  :⇔  ⋃{FISON}=⋃({FISON}\{F(k)})
 For each k ∈ ⋃{FISON}:  P(k)
 Because
none in linearly.ordered {FISON} is last.

None of the natural numbers is last.
How can Peano create the complete set by induction?
Regards, WM