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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 21 Jan 2025 13:44:50 +0100
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On 21.01.2025 13:17, Richard Damon wrote:
> On 1/21/25 6:45 AM, WM wrote:
>> 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.

> Which is a proof of ANY, not ALL together,

It is a proof of not any. The proof that not all together are necessary 
is this: U{F(1), F(2), F(3), ...} = U{F(2), F(3), F(4), ...}.

Regards, WM