Path: ...!news.roellig-ltd.de!open-news-network.org!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.math Subject: Re: How many different unit fractions are lessorequal than all unit fractions? Date: Wed, 9 Oct 2024 11:41:31 +0200 Organization: A noiseless patient Spider Lines: 53 Message-ID: References: <8ce3fac3a0c92d85c72fec966d424548baebe5af@i2pn2.org> <55cbb075e2f793e3c52f55af73c82c61d2ce8d44@i2pn2.org> <36fd2d02ead310af807440fc46d6987edbf8ef0e@i2pn2.org> <6a5f7390c3396e8eb407f3186ad5d7fce1c2224e@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 09 Oct 2024 11:41:32 +0200 (CEST) Injection-Info: dont-email.me; posting-host="600a0740231d71db0d3971ee5a02aa4d"; logging-data="2744687"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX185E6wpbWVBzJ1DM6woBv+JvyH9J9N8RRE=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:wkbaVg0aPj62n/ISq84kT7OLewA= In-Reply-To: <6a5f7390c3396e8eb407f3186ad5d7fce1c2224e@i2pn2.org> Content-Language: en-US Bytes: 4230 On 08.10.2024 21:17, joes wrote: > Am Tue, 08 Oct 2024 17:40:50 +0200 schrieb WM: >> On 08.10.2024 15:36, joes wrote: >>> Am Tue, 08 Oct 2024 12:40:26 +0200 schrieb WM: >>>> On 08.10.2024 12:04, Alan Mackenzie wrote: >>>>> WM wrote: >>> >>>>>> Hence all must be visible including the point next to zero, but they >>>>>> are not. >>>>> There is no point next to zero. >>>> Points either are or are not. The points that are include one point >>>> next to zero. >>> But not the point inbetween? >> If it exists then this point is next to zero. > Ah, then the former point wasn’t the one next to zero. Same goes for this > one. There are always infinitely many points between any two reals. > >>>> The infinite sets contain what? No natural numbers? Natural numbers >>>> dancing around, sometimes being in a set, sometimes not? An empty >>>> intersection requires that the infinite sets have different elements. >>> These are infinite sets: {2, 3, 4, …}, {3, 4, 5, …}, {4, 5, 6, …}. >>> They contain all naturals larger than a given one, and nothing else. >>> Every natural is part of a finite number of these sets (namely, its own >>> value is that number). The set {n+1, n+2, …} does not contain n and is >>> still infinite; there are (trivially) infinitely many further such >>> sets. All of them differ. >> All of them differ by a finite set of numbers (which is irrelevant) but >> contain an infinite set of numbers in common. > Every *finite* intersection. As long as infinitely many numbers are captivated in endsegments, only finitely many indices are available, and the intersection is between finitely many infinite endsegments. > Think about it this way: we are taking the limit of N\{0, 1, 2, …}. In the limit not a single natural number remains, let alone infinitely many. > >>>> Shrinking sets which remain infinite have not lost all elements. >>> This goes for every single of these sets, but not for their infinite(!) >>> intersection. >> If every single set is infinite, then the intersection is infinite too. >> These sets have lost some natural numbers but have kept infinitely many. > >>> If you imagine this as potential infinity, >> No, in potential infinity there are no endsegments. > Uh. So the naturals don’t have successors? They have successors but endsegments are sets and must be complete. Regards, WM