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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Fri, 31 Jan 2025 13:28:35 +0100
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On 31.01.2025 12:31, joes wrote:
> Am Thu, 30 Jan 2025 23:32:52 +0100 schrieb WM:
>> On 30.01.2025 15:30, Richard Damon wrote:
>>> On 1/30/25 4:14 AM, WM wrote:
>>>> On 29.01.2025 13:46, Richard Damon wrote:
>>>>
>>>>> We can in fact build an infinite set of infinite sets of FISONs whose
>>>>> union is the set of Natural Numbers,
>>>>
>> If there is a set with U(F(n)) = ℕ, then it has a first element that is
>> not completely useless.
> Not necessarily.

Necessarily, because otherwise all elements can be discarded.
> 

A sufficient
> set does not imply a necessary subset;

If there is no first element necessary, then all can be discarded, and 
the rest is empty - not sufficient to yield U(F(n)) = ℕ.
>> But all F(n) can be shown to be completely
>> useless because infinitely many natnumbers are missing.
> Again: every single one, or even an arbitrary finite number.
> If you have inf. many segments, you obviously have inf. many
> numbers.

All finite natural numbers as well as all FISONs obey the Peano axioms. 
Removing all leaves nothing, in particular no sufficient set for U(F(n)) 
= ℕ.

Regards, WM