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Path: ...!news.nobody.at!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> Newsgroups: sci.math Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness) Date: Fri, 20 Sep 2024 20:00:17 -0700 Organization: A noiseless patient Spider Lines: 126 Message-ID: <vclcs1$1eu11$1@dont-email.me> References: <vb4rde$22fb4$2@solani.org> <vc7lrp$2dd4k$1@dont-email.me> <jmudndRL4M71F3r7nZ2dnZfqn_ednZ2d@giganews.com> <a8b5743b-1bcf-49c1-936b-c4e92af9dec4@att.net> <3906cb72-4bad-4a2a-97c7-4da857adc7a4@att.net> <Q3udnQzBXvkGcnX7nZ2dnZfqn_QAAAAA@giganews.com> <05bf9c39-b715-41d6-b349-87ddc67941ff@att.net> <jpCcnchGB4mbTHT7nZ2dnZfqnPqdnZ2d@giganews.com> <1b54c6c9-8b85-4c59-865a-fb601eaf4e1f@att.net> <NHadnZwmjuswfnT7nZ2dnZfqnPGdnZ2d@giganews.com> <a9b31ddd-1084-45bd-bf18-1246f3d4dd1e@att.net> <vJadnS6Nw_pN3Hf7nZ2dnZfqnPcAAAAA@giganews.com> <vcfa5t$67r6$1@dont-email.me> <CsidnTV4T47-oHb7nZ2dnZfqnPoAAAAA@giganews.com> <8BycnZdkE-gTNXb7nZ2dnZfqnPednZ2d@giganews.com> <50cce993-5040-496a-822c-7f5d6558c22b@att.net> <vckdr6$1709n$2@dont-email.me> <67d492c9-5b13-404c-80a1-7aa0b70f12a6@att.net> <ndydnYYmD_VoeHD7nZ2dnZfqnPqdnZ2d@giganews.com> <ndydnYEmD_WAenD7nZ2dnZfqnPoAAAAA@giganews.com> <vckq03$18j12$1@dont-email.me> <ZSydnbsP2bUVr3P7nZ2dnZfqnPqdnZ2d@giganews.com> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 21 Sep 2024 05:00:17 +0200 (CEST) Injection-Info: dont-email.me; posting-host="9fabd7f027d8f87c91e752d958126e44"; logging-data="1538081"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/rez88IrHWdicgtfbmoowasOXkcOpqdrI=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:V88mo1pT37qVqP8rCtaF9ZgwWBo= In-Reply-To: <ZSydnbsP2bUVr3P7nZ2dnZfqnPqdnZ2d@giganews.com> Content-Language: en-US Bytes: 6195 On 9/20/2024 7:42 PM, Ross Finlayson wrote: > On 09/20/2024 02:38 PM, Chris M. Thomasson wrote: >> On 9/20/2024 2:20 PM, Ross Finlayson wrote: >>> On 09/20/2024 02:15 PM, Ross Finlayson wrote: >>>> On 09/20/2024 12:26 PM, Jim Burns wrote: >>>>> On 9/20/2024 2:10 PM, WM wrote: >>>>>> On 20.09.2024 19:51, Jim Burns wrote: >>>>> >>>>>>> Put pencil to paper and draw two curves which cross. >>>>>>> There is a point at which the curves intersect. >>>>>> >>>>>> This proves that no line has gaps. >>>>> >>>>> A point (hypothetically) next to 0 >>>>> has an absence of points between it and 0 >>>>> >>>>> No such absence of points exists. >>>>> No point which is next to 0 exists. >>>>> >>>>> >>>> >>>> Theorems or axioms? >>>> >>>> In the physics, think on your wave/particle duality, >>>> and the extended body, for example the wave-packet. >>>> >>>> "Drawing" a line, "tire en regle", or curve, >>>> has that when you put pencil to paper, >>>> and draw a line, or curve if you will, >>>> and life the pencil and put it back down, >>>> and draw another one, intersecting the first: >>>> the _curves_ cross. >>>> >>>> ... At a point, of for example where >>>> they're incident, they coincide. >>>> >>>> >>>> Then these lines-reals these iota-values >>>> are about the only "standard infinitesimals" >>>> there are: with extent you observe, density >>>> you observe, least-upper-bound as trivial, >>>> and measure as assigned, length assignment. >>>> >>>> >>> >>> Consider for example minutes between 1:00 and 3:00 >>> in this https://www.youtube.com/watch?v=fIexFF91Jl8 >>> "Moment and Motion: medical imaging technology". >>> >>> What it declares is that "complementary duals" have >>> that the points and the space and the space and the >>> points are for each other. >>> >>> >> >> Actually, each frame in this is part of a 3d process for constructing a >> volume that can be used in a volumetric renderer. It is close to medical >> imaging wrt DICOM: >> >> https://youtu.be/k9qpHcfiDho >> >> >> > > I sure had a head cold that day. > > Studying medical imaging that way was kind > of totem-ish, like, maybe if I study medical > imaging today, then sympathetic magic will > help kick my cold and diseases begin with 'C'. > Seemed helped, .... Yet, that's humor for you. > > > About "points-in-a-line", the classical notion > is "beads-on-a-string", here as with regards > to "the course-of-passage", that ordinals have > a course of passage through ordinals, and that > line-drawing, has a course of passage through > points. > > Divising a specialization of "in", "on", and > "about", the line, its points, helps to characterize > that line-drawing makes points "in" a line, that > zooming in via field operations points to, that > it establishes a vector field with all the arrows > reducing in region until their heads touch and > that they point-to, a point, "on" a line, then > as with regards to a super-fine comb each tooth > "about" the line. > > DICOM is a family of standards in medical imaging > data and metadata, sort of like JPEG for doctors. > See also "HL7". The JPEG2000 is pretty great, > I studied data formats a lot. A single DICOM file can contain the image stack that its based on, right? Fwiw, here is a render of some of my experiments: https://www.facebook.com/photo/?fbid=334035991088739&set=a.110008616824812 A screenshot: https://i.ibb.co/jGjycbN/image.png Strange results for sure. > > > Chaos theory and its terms is sort of folded into > "dynamical modeling", as with regards to for example > "limit cycles" for "attractors", then as with regards > to for example Nayfeh's tome on non-linear oscillations. > > > Anyways, here line-reals and signal-reals "exist" > next to field-reals, at least three models of > continuous domains, each with their own definition > of "continuous", all satisfying the same properties > of extent, density, completeness, and measure, from > which can be established most all their combined > character, as "real-valued". > > >