Re: The set of necessary FISONs

On 10.03.2025 23:36, Moebius wrote:

Am 10.03.2025 um 22:04 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

 

Z₀ is defined by induction: {} ∈ Z₀, and if {{{...{{{}}}...}}} with n
curly brackets ∈ Z₀ then {{{...{{{}}}...}}} with n+1 curly brackets ∈ Z₀.

 Actually, Z₀ is NOT "defined" (WM) by that.

It is ensured by that.
But in order to ensure the existence of "infinite" sets, we still need the following axiom, the essential content of which comes from Mr. Dedekind. ... The domain contains at least one set Z, which contains the zero set as an element and is such that each of its elements a corresponds to another element of the form {a} ... The set Z_0 contains the elements 0, {0}, {{0}}, etc. and may be referred to as a "number series", ... It forms the simplest example of a "countably infinite" set. [E. Zermelo: Investigations on the Foundations of Set Theory I, Mathematische Annalen (1908), p. 266]

But for Z₀ the following holds:
              {} ∈ Z₀
and
             Ax e Z₀: {x} e Z₀
             [Ax(x e Z₀ -> {x} e Z₀]
 We don't "count" "curly brackets" in this context.

We do. x e Z₀ implies that there are x+1 curly brackets.
And what please is the difference to the set F ensured by
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