Re: The set of necessary FISONs

On 31.01.2025 13:43, joes wrote:

Am Fri, 31 Jan 2025 11:21:58 +0100 schrieb WM:

On 31.01.2025 02:02, Richard Damon wrote:

On 1/30/25 2:28 PM, WM wrote:

>

But a set that has union ℕ is a set.

But since the "Set of Necessary FISONs" isn't actually a "Set"

take the set O of v. Neumann ordinals A(n) that you claim satisfies
U(A(n)) = ℕ.

It is a sufficient set. The necessary set is empty.

If no ordinal is necessary, then all can be discarded by induction. Then nothing remains. Nothing is not sufficient to obtain ℕ.

  From this set O every finite subset can be subtracted without changing
the result.

Key word „finite”.

All finite numbers are finite. None remain that cannot be discarded.

 

Therefore,  by induction, no finite A(n) remains.

Please formalise.

Peano axioms, namely n ==> n+1.
Regards, WM