Re: The set of necessary FISONs

Am 18.02.2025 um 19:22 schrieb Jim Burns:

On 2/18/2025 10:22 AM, WM wrote:

You (WM) have located a problem.
You try to work around it by not.mentioning it.
What you're not.mentioning is your assumption
that none of these sets are infinite.

Wrong. Induction has been invented for infinite sets.
Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden ... Axioms. [Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, S. 266]

⋃{F} = ℕ

Proof: If UF = ℕ is assumed, then F(1) can be omitted without changing the union of the remainder. And if F(n) can be omitted without changing this union, then also F(n+1) can be omitted without changing this union. That makes the omitted FISONs the inductive collection of all FISONs and proves the implication: If UF = ℕ, then { } = ℕ.
Regards, WM