Re: The set of necessary FISONs

On 11.03.2025 20:25, Jim Burns wrote:

On 3/11/2025 2:01 PM, WM wrote:

Zermelo defines Z to be an inductive set.

in order to ensure the existence of an infinite or inductive set.

Defining Z₀ to be inductive is insufficient
to prove, from a subset ⊆ Z₀ being inductive,
that the subset = the whole set Z₀

The set of numbers is the set Z₀. All other parts of Z are irrelevant.

 

|A| < ℵ₀  ∧  |B| = ℵ₀  ⇒  |B\A| = |B| = ℵ₀

>
|Z₀| < ℵ₀

 No.

Take Cantor's ℕ. Delete 1. If you have deleted n, delete n+1. In all steps ℵ₀ numbers remain.

Z₀ is inductive.

Take Cantor's ℕ denoted as { }, {{ }}, {{{ }}}, ... . Delete { }. If you have deleted n brackets, delete n+1 brackets. In all steps ℵ₀ elements remain.
Regards, WM