Re: The set of necessary FISONs

On 09.02.2025 01:00, Jim Burns wrote:

On 2/8/2025 4:54 PM, WM wrote:

The axiom of induction says:
If any property or predicate P satisfies
(P(1) /\ ∀k(P(k) ==> P(k+1)),
then it describes all elements of an inductive = infinite set.

 Not for all inductive sets.
For all minimal.inductive sets.

May be called so.

 minimal.inductive ≠ inductive ≠ infinite

Then you are wrong. Every inductive set is infinite.

 

That is satisfied by the set M of all FISONs which are useless in U(A(n)) = ℕ.

 Without exception,
the union of FISONs.after is ℕ

No.

Therefore U(F(n)) = ℕ ==>
U{ } = { } = ℕ.

 Why that '==>' ?

 From the assumption U(F(n)) = ℕ I have derived that { } = ℕ.

 My best guess at why you claim  U{ } = { } = ℕ
is that you (WM) are assuming that,
for some FISON (ie, F(ω-1)) such that
there are no FISONs.after.

No.

 You (WM) haven't given any other reason.

If U(F(n)) = ℕ, then F(1) can be omitted without changing the result. If F(k) can be omitted, then F(k+1) can be omitted too. The set of FISONs which can be omitted is an inductive set, i.e., all FISONs.
Regards, WM