Re: The set of necessary FISONs

On 07.03.2025 00:23, Jim Burns wrote:

On 3/6/2025 12:15 PM, WM wrote:

Constructions[proofs] construct[prove.to.exist]
infinite sets,
but those constructions[proofs] aren't
endless iterations.

The proof is finite, few lines. The method used is infinite = endless, never leaving the finite domain.

This proof is finite:
⎛ Infinity: inductive Z exists.
⎜ PowerSet: 𝒫(Z) exists.
⎜ Separation: 𝒫ⁱⁿᵈ(Z) exists.
⎝ Separation: ⋂𝒫ⁱⁿᵈ(Z) exists.

This proof is finite too: { } ∈ Z₀, and if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀ then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀. But is does not cover or reach any number larger than all finite numbers.

We know Z₀ exists

Not without the induction.

Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden ... Axioms.

 Those axioms ensure the existence of Z₀
but not the way that you (WM) think they ensure it.

You are wrong, and you know it.

 

[Zermelo: Untersuchungen über die Grundlagen
der Mengenlehre I, S. 266]
The elements are defined by induction
in order to guarantee the existence of infinite sets.

That is what I obtain from Zermelo.
Regards, WM