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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 11 Mar 2025 22:28:53 +0100
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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