Re: The set of necessary FISONs

On 15.02.2025 18:38, joes wrote:

Am Sat, 15 Feb 2025 15:55:55 +0100 schrieb WM:

On 15.02.2025 14:40, Richard Damon wrote:
>

And since you build your claimed "set" by Naive Set theory, your sets
are worthless.

Zermelo constructs his set in modern set theory. Induction concerns the
whole set. Compare Zermelo: "In order to secure the existence of
infinite sets, we need the following axiom." [Zermelo: Untersuchungen
über die Grundlagen der Mengenlehre I, S. 266] This is the axiom of
infinity proved by induction. It ascertains the existence of an infinite
set. It ascertains the set Z, Z_0 and the union of singletons ℕ.

>
The set you show is empty, is the set of FISONs that are individually
not necessary.

The remaining set should have at least one element.

Indeed, if you remove any number of FISONs, infinitely many remain.

Induction proves an inductive set F of FISONs that can be removed without changing the result U(F) = ℕ ==> U(F\F) = ℕ.
Regards, WM