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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: The set of necessary FISONs
Date: Tue, 21 Jan 2025 12:45:01 +0100
Organization: A noiseless patient Spider
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All finite initial segments of natural numbers, FISONs F(n) = {1, 2, 3, 
...., n} as well as their union are less than the set ℕ of natural numbers.

Proof: Assume UF(n) = ℕ. The small FISONs are not necessary. What is the 
first necessary FISON? There is none! All can be dropped. But according 
to Cantor's Theorem B, every non-empty set of different numbers of the 
first and the second number class has a smallest number, a minimum. This 
proves that the set of indices n of necessary F(n), by not having a 
first element, is empty.

Regards, WM