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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Fri, 31 Jan 2025 11:31:23 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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. You may add the smallest segment that is not in the
set to be unified (if the set does not include all). A sufficient
set does not imply a necessary subset; indeed, a necessary set may
not exist at all. It would require that a number existed in only one
segment; however every segment is a subset of the successors.

> 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.

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.