Re: The set of necessary FISONs

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

On 06.03.2025 14:47, Jim Burns wrote:

On 3/6/2025 4:15 AM, WM wrote:

Am 06.03.2025 um 10:06 schrieb joes:

Am Wed, 05 Mar 2025 22:05:04 +0100 schrieb WM:

Therefore iteration fails to produce
actual infinity.

>
As an element, but not as
the number of elements (=the size of the set).

>
We do NOT construct[make] sets.
We construct[know] sets.

>
Without their construction/proof
we don't know whether infinite sets exist at all.

Constructions[proofs] construct[prove.to.exist]
infinite sets,
but those constructions[proofs] aren't
endless iterations.
A proof needs to be finite (not endless).
However,
a proof doesn't need to be about something finite.
This proof is finite:
⎛ Infinity: inductive Z exists.
⎜ PowerSet: 𝒫(Z) exists.
⎜ Separation: 𝒫ⁱⁿᵈ(Z) exists.
⎝ Separation: ⋂𝒫ⁱⁿᵈ(Z) exists.
What it constructs[proves.to.exist], ⋂𝒫ⁱⁿᵈ(Z),
is infinite.
⋂𝒫ⁱⁿᵈ(Z) = {{},{{}},{{{}}},...}
⋂𝒫ⁱⁿᵈ(Z) = Z₀ [emptiest inductive]
We know Z₀ exists and is suitable for
a proof.by.induction by virtue of
claims being in a
finite sequences of finite.length claims
(each true.or.not.first.false).
That is what a proof is,
not an infinite iteration.

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.

[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.