Re: The set of necessary FISONs

Am 19.02.2025 um 15:50 schrieb joes:

Am Wed, 19 Feb 2025 15:31:54 +0100 schrieb WM:

They didn’t say that by induction, you can infer properties of a set
from its elements - like being finite or not. Compare {Q, N, R},
the set of endsegments, and {0, 1, 2}.

By induction all elements can be defined. This guarantees the existence of an infinite set. Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden ... Axioms. [Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, S. 266]

„If every natural is finite, then there are only finitely many
naturals.”?

Word on this?

The set of all natural numbers which can be defined is (potentially in-) finite. infinitesimally smaller than 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