Re: The set of necessary FISONs

On 07.02.2025 17:10, Jim Burns wrote:

On 2/6/2025 2:32 PM, WM wrote:

I prefer Wikipedia:
∀P (P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)).

>
That's intended to be part of the definition of ℕ₁

>
As well it is
the definition of the collection of all FISONs.

 Which is curious, when one considers that
the collection of all FISONs appears nowhere in it.

The axiom of induction says: If any property or predicate P satifies
(P(1) /\ ∀k(P(k) ==> P(k+1)), then it describes all elements of an inductive = infinite set. That is satisfied by the set M of all FISONs which are useless in U(A(n)) = ℕ.

If the set M is described as the smallest set satisfying
F(1) ∈ M and F(n) ∈ M ==> F(n+1) ∈ M
then M contains
all FISONs which can be subtracted from U(Fn))
without changing the assumed result ℕ.

 Also, any superset of (emptiest) M
contains at least what M contains, and
thus also contains all FISONs.

But only all FISONs can be discarded.
Regards, WM