Re: The set of necessary FISONs

On 11.03.2025 23:51, Jim Burns wrote:

On 3/11/2025 5:28 PM, WM wrote:

On 11.03.2025 20:25, Jim Burns wrote:

On 3/11/2025 2:01 PM, WM wrote:

 

Zermelo defines Z to be an inductive set.

>
in order to ensure the existence of
an infinite or inductive set.

 No.

Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden ... Axioms. [Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, S. 266]

There is a definition and there is an axiom.

Where are they? Please quote a definition which is outside of the axiom.

They are different kinds of things.

And you are unable to understand the connection between these two things. Is it the language? Or is it a general deficit?

 The definition of Z states that
Z is inductive.

Please quote this definition which is not the axiom.

 

in order to ensure the existence of
an infinite or inductive set.

 A definition doesn't ensure existence.
That's what an axiom does.

This axiom ensures the sequence of numbers by the same induction which ensures the dark numbers:
{} ∈ Z₀, and for all x: if x ∈ Z₀ then {x} ∈ Z₀
ℕ \ F(1) = ℵo,
and if
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n) = ℵo
then
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n+1) = ℵo.
Regards, WM