Re: The set of necessary FISONs

On 08.03.2025 12:58, Jim Burns wrote:

On 3/8/2025 3:45 AM, WM wrote:

You need not the intersection however
because
Z₀ can also be defined by
{ } ∈ Z₀, and
if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀.

Zermelo's Axiom of Infinity describes Z
Z ∋ {} ∧ ∀a: Z ∋ a ⇒ Z ∋ {a}
Multiple sets satisfy that unique description.
That's an indefinite description.

My description is definite.

If only
{{{...{{{ }}}...}}} with finitely.many curly brackets
each with immediate.predecessor ⋃{{{...{{{ }}}...}}}
and with {} being one of its priors
are in Z₀
then, yes, that is a definite description.
 And, yes, that seems to be what you meant.
If you ever want to make your descriptions clearer,
you won't be getting complaints from me.

That is exactly what I meant. And that's also the induction of my argument UF = ℕ  ==> Ø = ℕ.

 

I suspect it can be said,
but I don't know how, right now,
and I'm sure that you (WM) haven't said how.

>
Is it so bewildering to
replace "1" and "if n ∈ ℕ then n+1 ∈ ℕ"
by Zermelo's notation?

 If n ranges over transfinite ordinals,
then your ℕ is unreliable in proof.by.induction.
It's not what Zermelo, von Neumann, or Cantor
mean by ℕ
 If n ranges over only finite ordinals,
then
you didn't explicitly say that,

Induction ranges over finite ordinals.

  which explicitness is the reason for writing,
and,
if you had explicitly said that,
you should have said somewhere
what a finite ordinal is,

That is not under discussion here. n is usually denoting a natural number. FISONs are finite by definition.
Regards, WM