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Path: ...!news.mixmin.net!weretis.net!feeder8.news.weretis.net!pasdenom.info!from-devjntp Message-ID: <l90BjXTiETgjHOXXIVlMxb7MmW4@jntp> JNTP-Route: news2.nemoweb.net JNTP-DataType: Article Subject: Re: Replacement of Cardinality References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp> <1b259a91952c93a56ad1e0063a2d7440aed185f2@i2pn2.org> <rHIaB-dFODVqSY7-aRnf4ItTyG0@jntp> <20e0e340532aa10bcc86e51eb5d19d006acefb12@i2pn2.org> <el_h_RPLN1ZVr_KeaLK-R-0CPpY@jntp> <411a6d693f5dac5ec7cf51a239a9570ac5ce7bd4@i2pn2.org> <9ynqddb0aIxRIgzLd8YZuWqFJQQ@jntp> <e12da78fd53e95ae8b54c4b0f100c10542acfa1a@i2pn2.org> <-e7H52W39rR5zxvT-LWQK-OpG5Y@jntp> <e7f793fb1fd82c90442679bfbcd2a334f10b7b13@i2pn2.org> Newsgroups: sci.logic,sci.math JNTP-HashClient: 1hgiQxC5hZXNgHbCxa6Y1habYmk JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw JNTP-Uri: http://news2.nemoweb.net/?DataID=l90BjXTiETgjHOXXIVlMxb7MmW4@jntp User-Agent: Nemo/0.999a JNTP-OriginServer: news2.nemoweb.net Date: Fri, 02 Aug 24 16:32:50 +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="2024-08-02T16:32:50Z/8973198"; 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: 2800 Lines: 33 Le 02/08/2024 à 18:24, Richard Damon a écrit : > On 8/2/24 12:05 PM, WM wrote: > The claim is that there is a finite difference that seperates any two > specific unit fractions, Of course. > > Not that there is a single finite difference that seperates any two > arbirtry unit fractions. Between any two arbitrary unit fractions there are ℵ₀*2ℵ₀ points. > > Not understanding the order of the arguement Between any two arbitrary unit fractions there are ℵ₀*2ℵ₀ points. No order necessary. > > For your example, for any x, there is an eps = 1/(ceil(1/x)+1) that > provides a unit fraction smaller than x, No, eps has to be chosen. x has to be chosen. My claim is that most are dark and cannot be chosen. Note: For every 1/n there exists a smaller real number. But they cannot be chosen because most 1/n already cannot be chosen. Proof: For every chosen eps you fail to separate infinitely many unit fractions. > and thus NUF(x) can not be 1 > for any finite x. But for x belonging to the first ℵ₀*2ℵ₀ points we have less than ℵ₀ unit fractions. Regards, WM