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Path: ...!weretis.net!feeder8.news.weretis.net!pasdenom.info!from-devjntp Message-ID: <6n5Etd_3NSOCHqlQ64GwbTcNa5w@jntp> JNTP-Route: news2.nemoweb.net JNTP-DataType: Article Subject: Re: how References: <qHqKnNhkFFpow5Tl3Eiz12-8JEI@jntp> <86943ca8-a023-46d1-b068-78a03a0b0e76@att.net> <psHXzIfyki-gj3aETK8ifAexIXU@jntp> <b08a9d92-3824-4004-b114-4b560047c363@att.net> <3Q7-Zmb9wCOxkubzUbgi5US3BXA@jntp> <72cb87f2-3978-41e7-b460-911d7f593179@att.net> <L45dAOJHQfxivFfY5S_SAJaT-R8@jntp> <97424d4d-afe6-4f14-b144-970164bdab3d@att.net> <Nmd47wYK51V9jMW5YOVNtW21cGs@jntp> <61b4043f-cf4d-4449-b323-94780954af3f@att.net> Newsgroups: sci.math JNTP-HashClient: zmsdzwTFLMn6Jrs-N9LTGnJpEcA JNTP-ThreadID: 4YLc1knY-8u5i_KQ0oWqy89D7aY JNTP-Uri: http://news2.nemoweb.net/?DataID=6n5Etd_3NSOCHqlQ64GwbTcNa5w@jntp User-Agent: Nemo/0.999a JNTP-OriginServer: news2.nemoweb.net Date: Mon, 24 Jun 24 18:41:31 +0000 Organization: Nemoweb JNTP-Browser: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/126.0.0.0 Safari/537.36 Injection-Info: news2.nemoweb.net; posting-host="25d5a506365fc8262443ce1bd287e5d0233c1bef"; logging-data="2024-06-24T18:41:31Z/8916809"; posting-account="217@news2.nemoweb.net"; mail-complaints-to="julien.arlandis@gmail.com" JNTP-ProtocolVersion: 0.21.1 JNTP-Server: PhpNemoServer/0.94.5 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit X-JNTP-JsonNewsGateway: 0.96 From: WM <wolfgang.mueckenheim@tha.de> Bytes: 4460 Lines: 102 Le 24/06/2024 à 19:18, Jim Burns a écrit : > On 6/23/2024 4:04 PM, WM wrote: >> Le 23/06/2024 à 19:58, Jim Burns a écrit : > >>> U{FISON} is the union of the set of >>> both necessary and unnecessary FISONs >> >> The unnecessary FISONs can be removed. > > Your definition of 'unnecessary' is that > an unnecessaryᵂᴹ FISON when removed > leaves the union of remaining FISONs unchanged. Yes. And that can be decided for every FISON. > One unnecessaryᵂᴹ FISON can be removed. > One. Every one. By induction. > > {FISON} is inductive and well.ordered. Yes. Every one can be removed without changing the union. Potential infinity. > > Because well.ordered, > Any necessaryᵂᴹ FISON in {FISON} > is last in {FISON} Yes. But there is no last one in a potentially infinite sequence. With F(n) also F(n^n) is present. There are oo many bright FISONs. > > Because inductive, > No FISON is last in {FISON} But every FISON has oo bright and ℵo dark successors. > > Any _one_ FISON can be removed and > leave the union of remaining FISONs unchanged. > Any _one_ FISON is unnecessaryᵂᴹ > > Removing all of those which > singly don't change the union > changes the union, ⋃{}≠ℕ > It is not a contradiction. The contradiction is this: For every FISON we can determine whether it is necessary. Therefore the collection of neccessary FISONs is well-defined and, if being a set, has a smallest element. If its union could be ℕ, the necessary FISONs must be a set and have a smallest element. That is not the case. > > Compare to: > Suppose I have 5 elephants, and > an unnecessaryᴶᴮ elephant is one when removed > leaves 4 elephants. > > Any _one_ elephant can be removed and > leave 4 elephants. > Any _one_ elephant is unnecessaryᴶᴮ First enumerate them. Then remove 1, 2, 3, 4. Stop. > > Removing all which > singly leave 4 > doesn't leave 4 > It is not a contradiction. > It is arithmetic. Don't evade. Apply the logic: If the union of FISONs is ℕ, then all unnecessary FISONs can be dropped and the remaining FISONs must be a set with union ℕ. If they are a set they have a smallest element. That is not the case. Hence they are not a set. > >>>> The union cannot be larger than all its FISONs. >>> >>> The union cannot be smaller than any FISON. >> >> Agreed. > >>> Each FISON is smaller than another FISON, > >> Therefore the union is a FISON. > > Therefore the union is NOT a FISON. Observe logic. The union cannot be larger than all its FISONs. The union cannot be smaller than any FISON. Not larger and not smaller means ...? >> It runs through ℕ >> but never covers a substantial part. > > Related: > ∀ᴿx>0: NUF(x) = ℵ₀ Yes, that is as much violating logic. But I do not accept that. Regards, WM