Path: ...!feeds.phibee-telecom.net!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness) Date: Sun, 15 Sep 2024 15:18:01 -0700 Organization: A noiseless patient Spider Lines: 117 Message-ID: References: <405557f7289631d63264c712d137244c940b9926@i2pn2.org> <13c08e96ad635f8142b38d89863a80caf17a32a8@i2pn2.org> <4faa63d0ff8c163f01a38736aeb5732184218a29@i2pn2.org> <7-ycnVjnAIKXynr7nZ2dnZfqn_qdnZ2d@giganews.com> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 16 Sep 2024 00:18:01 +0200 (CEST) Injection-Info: dont-email.me; posting-host="0f36da695d9f6b2ce32e431416f90f90"; logging-data="2536381"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19Oic81FGZNllmUxqcku5cM1CfYIrTPbLs=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:tTrCD1Eq7AjtOKfFDT4N90D15cw= Content-Language: en-US In-Reply-To: Bytes: 6147 On 9/15/2024 3:07 PM, FromTheRafters wrote: > on 9/15/2024, Ross Finlayson supposed : >> On 09/15/2024 11:03 AM, FromTheRafters wrote: >>> After serious thinking Ross Finlayson wrote : >>>> On 09/14/2024 09:27 AM, Ross Finlayson wrote: >>>>> On 09/13/2024 04:05 PM, FromTheRafters wrote: >>>>>> WM explained : >>>>>>> On 13.09.2024 17:52, Richard Damon wrote: >>>>>>>> On 9/13/24 11:41 AM, WM wrote: >>>>>>> >>>>>>>>> Between [0, 1] and (0, 1] there is nothing, there is not a spot or >>>>>>>>> point of the interval. >>>>>>>>> >>>>>>>> But that doesn't mean there is a lowest most point in (0, 1] as any >>>>>>>> point you might want to call it will have another point between  it >>>>>>>> and 0. >>>>>>> >>>>>>> I will not call any point but consider all points. There is no point >>>>>>> smaller than all points in the open interval but a smallest one. >>>>>>> Only >>>>>>> 0 is smaller than all. >>>>>>>> >>>>>>>> Note, I said between the point your THINK is the first, there is no >>>>>>>> such point, and thus you are agreeing to that fact. >>>>>>>> >>>>>>>> You can only have a first point in the open interval if the >>>>>>>> interval >>>>>>>> has only a finite number of points, >>>>>>> >>>>>>> No, that is your big mistake. In the interval [0, 1] there is a >>>>>>> point >>>>>>> next to 0 and a point next to 1, and infinitely many are beteen >>>>>>> them. >>>>>> >>>>>> Define 'next' in this context. >>>>> >>>>> The context is "continuous domains", >>>>> there are multiple models of continuous domains, >>>>> one of them is "iota-values" or "line-reals", >>>>> which is a model of a contiguity so fine as >>>>> a model of continuity, where it's, "EF(1)". >>>>> >>>>> Of course, the models of continuous domains are >>>>> distinct as with regards to their definitions of >>>>> continuity and completeness of operations, so >>>>> it entails a bit of book-keeping to keep things. >>>>> >>>>> Oh, you don't have one of those, ..., well, you >>>>> can always look to Aristotle, who has at least >>>>> two, and Zeno's always looking for how to arrive >>>>> at not being a fool, then fast-forward to Bishop >>>>> and Cheng who constructively go about making it >>>>> so, and for topology there's Vickers who helps >>>>> reflect that in topology there are various topologies >>>>> not necessarily the standard open topology, in case >>>>> you're thorough about these matters and want to >>>>> help square away various models of continuity, >>>>> continuous domains, continuous topologies their >>>>> own first and final, Cantor space, and law(s) of >>>>> large numbers. >>>>> >>>>> >>>> >>>> "What, no witty rejoinder?" >>> >>> What you said has no relation to the 'nextness' of elements in discrete >>> sets. What is 'next' to Pi+2 in the reals? >> >> In the, "hyper-reals", it's its neighbors, >> in the line-reals, put's previous and next, >> in the field-reals, there's none, >> and in the signal-reals, there's nothing. >> >> Or, you know, "noise". >> >> Of course the hyper-reals are said to be only >> a "conservative" meaning "meaningless" extension >> to the standards, so, only the line-reals say >> anything about it at all. >> >> Except nothing, .... >> >> >> I wonder what you think of something like Hilbert's >> "postulate of continuity" for geometry, as with >> regards to that in the course-of-passage of >> the growth of a continuous quantity, it encounters, >> in order, each of the points in the line. >> >> Just ignore it? > > What is the successor function on the reals? Give me that, and maybe we > can find the 'next' number greater than Pi. define a granularity and you can explore some numbers. However this is finite and misses infinite points. They are not countable and are infinitely dense. This is just a finite view based on a given granularity: // kind of small... ;^) granularity = (10^(-696), 0, 0) p0 = (-1, 0, 0) p1 = (1, 0, 0) pdif = p1 - p0; cur = p0; // start at cur and go off into infinity and beyond! ;^) for (;;) { next = cur + granularity plot_line(cur, next); cur = next; } ;^)