Path: news.eternal-september.org!eternal-september.org!feeder3.eternal-september.org!i2pn.org!rocksolid2!i2pn2.org!.POSTED!not-for-mail
From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sat, 25 Jan 2025 13:56:49 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
Message-ID: <965ee9868f28953d437113d0d38f069815499a3f@i2pn2.org>
References: <vmo1bs$1rnl$1@dont-email.me>
	<ee4e189d-56bc-4ed7-b7b7-cd71fc4d2349@att.net> <vmqh9k$urua$1@dont-email.me>
	<903de8e1-3538-4cfe-9f7a-6509eda47ab6@att.net>
	<vmsvfv$1hn5k$1@dont-email.me>
	<3844edd7-0750-4418-bff6-2759817446b3@att.net>
	<vmvn1h$25r19$1@dont-email.me>
	<27377646-137a-4f8f-a7bb-a75707b2da96@att.net>
	<vn2gcf$2ouuo$2@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
Injection-Date: Sat, 25 Jan 2025 13:56:49 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="1077292"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="nS1KMHaUuWOnF/ukOJzx6Ssd8y16q9UPs1GZ+I3D0CM";
User-Agent: Pan/0.145 (Duplicitous mercenary valetism; d7e168a
 git.gnome.org/pan2)
X-Spam-Checker-Version: SpamAssassin 4.0.0

Am Sat, 25 Jan 2025 12:02:39 +0100 schrieb WM:
> On 24.01.2025 16:44, Jim Burns wrote:
>> On 1/24/2025 4:37 AM, WM wrote:
>>> On 23.01.2025 16:18, Jim Burns wrote:
>> 
>> The union of all FISONs covers UF(n)
> 
> Simply contradicted by:
> ∀n ∈ UF(n): |ℕ \ {1, 2, 3, ..., n}| = ℵo 
[assuming UF(n) means {1, 2, ..., n}]
That just says that N is not a single FISON.

>> Each FISON is a proper subset of another FISON. Each FISON is a proper
>> subset of UF(n). No FISON is UF(n)
> That is potential infinity.
No, it’s just a factual description.

>> Whatever contains each FISON  contains UF(n)
> Alas this is not a set but a (potentially in-) finite changing
> collection.
No. Can you not conceive of an infinite set?

>>> Otherwise Cantor's theorem would require the existence of a first
>>> necessary FISON.
There is obviously no single FISON that is equal to N.

> Do you agree that or every FISON the question whether it is necessary
> can be answered?
Yes, in the negative. That does not imply the nonexistence of a
sufficient set.

>>> Up to every FISON |ℕ \ {1, 2, 3, ..., n}| = ℵo.
N is not a FISON.

>> For any two FISONs {1,2,...,j} {1,2,...,k} their sum
>> {1,2,...,j,j+1,j+2,...,j+k} is a FISON
> Therefore the union cannot be larger than a FISON.
The union is infinite, since each FISON adds a different number.

> The infinite union is the infinite FISON. But there is no infinite FISON
The limit of FISONs is N.

>> ⎛ Consider Bob such that,
>> ⎜ before all FISON.end.swaps n⇄n+1 ⎜ Bob is in the first FISON.end 0 ⎜
>> ⎜ If Bob is in FISON.end n ⎜ then ⎜ it is after n-1⇄n and before n⇄n+1
>> ⎜ If it is after all FISON.end.swaps ⎜ then Bob is not.in any
>> FISON.end,
>> ⎜ even though ⎜ no FISON.end.swap takes Bob ⎝ anywhere else.
>> 
> Swaps cannot eliminate Bob. He remains but i the darkness.
An infinite sequence of swaps may not correspond to a single swap.

-- 
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.