Path: ...!weretis.net!feeder8.news.weretis.net!pasdenom.info!from-devjntp Message-ID: JNTP-Route: news2.nemoweb.net JNTP-DataType: Article Subject: Re: Replacement of Cardinality References: <155cdc8a628d47be1632791227bccf99425b1d5e@i2pn2.org> <227e12c2862e139d022279d3ae5bdd34427bafae@i2pn2.org> Newsgroups: sci.logic,sci.math JNTP-HashClient: 0mN-0hVYN899sRjTRyNhOtcOav0 JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw JNTP-Uri: http://news2.nemoweb.net/?DataID=JWO2G07WD4l3OOZ0iu5_ESbFkKs@jntp User-Agent: Nemo/0.999a JNTP-OriginServer: news2.nemoweb.net Date: Wed, 21 Aug 24 12:32:30 +0000 Organization: Nemoweb JNTP-Browser: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/127.0.0.0 Safari/537.36 Injection-Info: news2.nemoweb.net; posting-host="82b75c1d0a83e677ff646b52485f72f8b23749df"; logging-data="2024-08-21T12:32:30Z/8995089"; 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 Bytes: 2929 Lines: 33 Le 21/08/2024 à 13:32, Richard Damon a écrit : > On 8/21/24 6:44 AM, WM wrote: >> Le 20/08/2024 à 23:25, FromTheRafters a écrit : >>> WM explained : >>>> Le 20/08/2024 à 12:31, FromTheRafters a écrit : >>>>> on 8/19/2024, Richard Damon supposed : >>>> >>>>>> You can not derive a first number > 0 in any of the Number System >>>>>> that we have been talking about, Unit Fractions, Rationals or >>>>>> Reals, so you can't claim it to exist. >>>>> >>>>> Not in their natural ordering. >>>> >>>> Dark numbers have no discernible order. It is impossible to find the >>>> smallest unit fraction or the next one or the next one. It is only >>>> possible to prove that NUF(x) grows by 1 at every unit fraction. It >>>> starts from 0. >> >>> Normally, the unit fractions are listed in the sequence one over one, >>> one over two, one over three etcetera. There is a first but no last. >>> Now you have started from the wrong 'end' >> >> No, I have started from the other end. It exists at x > 0 because NUF(0) >> = 0. > But the other end doesn't "begin" with a first Natural Number Unit > fraction, if it has a beginning that will be a trans-finite number. No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all and only reciprocals of natural numbers. Regards, WM