Re: The set of necessary FISONs

On 2/22/25 1:02 PM, WM wrote:

On 22.02.2025 15:03, Richard Damon wrote:

On 2/22/25 7:35 AM, WM wrote:

The inductive step is proved in my example from |ℕ \ {1, 2, 3, ..., n}| = ℵo, i.e., by several axioms, and in Zermelo's example by a single axiom.

 

Which just proves that no specific FISON is individually REQURED to make the set of Natural Numbers.

 If every FISON can be omitted, ten nothing remains for a sufficient set. Because if there was any sufficient set, it would have a first FISON.
 Regards, WM

The fact that none of them are individually requred doesn't mean you can't use them.
I guess you do think that we can't factor 36, since none of the factors are requires, so all can be omitted from the set you can use.
Yes, there is a first FISON in a sufficient set, that FISON is { 1 }, as part of the set { {1}, {1, 2}, {1, 2, 3}, ...}
You still have the meaning of the words incorrect.
A sufficient set can contain elements that are not needed.
I think you mean a necessary set, but the problem is there is no requirements of the existance of a necessary set, just a sufficient set.