Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Wed, 25 Sep 2024 18:44:27 +0000 Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness) Newsgroups: sci.math References: <3906cb72-4bad-4a2a-97c7-4da857adc7a4@att.net> <05bf9c39-b715-41d6-b349-87ddc67941ff@att.net> <1b54c6c9-8b85-4c59-865a-fb601eaf4e1f@att.net> <8BycnZdkE-gTNXb7nZ2dnZfqnPednZ2d@giganews.com> <50cce993-5040-496a-822c-7f5d6558c22b@att.net> <67d492c9-5b13-404c-80a1-7aa0b70f12a6@att.net> <9b2ffafe-78a1-4854-a27c-362a8d3a3552@att.net> From: Ross Finlayson Date: Wed, 25 Sep 2024 11:44:22 -0700 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: <9b2ffafe-78a1-4854-a27c-362a8d3a3552@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 267 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-Sie532IPV6cVdBKjQOcwMCHThsS5LIx+rHrV9G709AsM32bSNTAkQhBTkySSdv60AIrGuSURXS7w4/8!hlhjDZVLjBXXeWHw8wijvPT/ao8tj8J6L30YZF2uohmJGB2iBcSOWjXY/5/9Xrlhaz+U0ygEMF1U X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 12865 On 09/25/2024 10:11 AM, Jim Burns wrote: > On 9/24/2024 11:03 PM, Ross Finlayson wrote: >> On 09/24/2024 02:47 PM, Ross Finlayson wrote: >>> On 09/24/2024 01:35 PM, Ross Finlayson wrote: >>>> On 09/24/2024 10:22 AM, Jim Burns wrote: >>>>> On 9/20/2024 5:15 PM, Ross Finlayson wrote: > >>>>>> 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. >>>>> >>>>> Lines with the least.upper.bound property >>>>> (equivalent to "crossing must intersect") >>>>> do not have infinitesimals. >>>>> >>>>> For example, >>>>> there are no infinitesimals >>>>> between 0 and all the _finite_ unit.fractions. >>>>> >>>>> ⎛ Each positive point has >>>>> ⎜ a finite unit.fraction between it and 0 >>>>> ⎜ >>>>> ⎜⎛ Otherwise, >>>>> ⎜⎜ greatest.lower.bound β of finite unit.fractions >>>>> ⎜⎜ is positive, and >>>>> ⎜⎜ not.bounding 2⋅β > finite ⅟k >>>>> ⎜⎜ ½⋅β > ¼⋅⅟k >>>>> ⎜⎜ β > ½⋅β > ¼⋅⅟k >>>>> ⎜⎜ greatest.lower.bound β is not.bounding, >>>>> ⎝⎝ which is gibberish. >>>> >>>> Well now, there are as many kinds infinitesimals >>>> as there are infinities, > > In this discussion, by 'infinitesimal', I mean > a point δ between 0 and all finite.unit.fractions. > infinitesimal δ :⇔ > ∀k ∈ ℕ: 0 < δ < ⅟k ⇔ > 0 < δ ≤ᵉᵃᶜʰ ⅟ℕ > > β = greatest.lower.bound of finite.unit.fractions > ∀ᴿr: r ≤ᵉᵃᶜʰ ⅟ℕ ⇒ r ≤ β ≤ᵉᵃᶜʰ ⅟ℕ > > ℕ is well.ordered and (≠0) nexted > ∀S ⊆ ℕ: S={} ∨ ∃k ∈ S: k ≤ᵉᵃᶜʰ S > ∀j ∈ ℕ: ∃!k ∈ ℕ\{0}: j+1=k > ∀k ∈ ℕ\{0}: ∃!j ∈ ℕ: j+1=k > ∀j ∈ ℕ: j < j+1 ∧ ¬∃k ∈ ℕ: j < k < j+1 > > lemma: > ¬∃δ: 0 < δ ≤ᵉᵃᶜʰ ⅟ℕ > There are no infinitesimals like that. > > My current purpose is to dissuade > believers in a smallest unit.fraction > from believing in a smallest unit.fraction. > > Do you (RF) think that other systems of infinitesimals > might be of use in that discussion? > If you think so, why do you? > >>>> and all in a general sense differing in >>>> differences quite clustered about zero, >>>> make for that Peano, Dodgson, Veronese, >>>> Stolz, Leibniz, MacLaurin, Price, >>>> the entire field of infinitesimal analysis as >>>> what real analysis was named for hundreds of years, >>>> make for that even Robinson's >>>> rather modest and of no analytical character >>>> the hyper-reals, or >>>> as among Conway's surreal numbers, >>>> has that most people's ideas of infinitesimals >>>> are exactly as an infinite of them in [0,1], >>>> constant monotone strictly increasing, >>>> as with regards to "asymptotic equipartitioning" >>>> and other aspects of higher, and lower, mathematics. >>>> >>>> Newton's "fluxions", Aristotle's contemplations and >>>> deliberations about atoms, Zeno's classical expositions, >>>> quite a few of these have infinitesimals all quite >>>> throughout every region of the linear continuum. >>>> >>>> Maybe Hardy's pure mathematics makes for conflating >>>> the objects of geometry, points and lines, with >>>> a descriptive set theory's, a theory with only >>>> one relation and only one-way, point-sets, yet >>>> for making a theory with them all together, >>>> makes for that since antiquity and through >>>> today, notions like Bell's smooth analysis, >>>> and Nelson's Internal Set Theory, if you >>>> didn't know, each have that along the linear >>>> continuum: are not "not infinitesimals". >>>> >>>> Here these "iota-values" are considered >>>> "standard infinitesimals". >>>> >>>> Then, in the complete ordered field, >>>> there's nothing to say about them >>>> except nothing, well, some have that >>>> its properties of least-upper-bound >>>> and measure are actually courtesy already >>>> a more fundamental continuum, in the theory, >>>> as a constant, and not just stipulated >>>> to match expectations. >>>> >>>> The MacLaurin's infinitesimals and then for >>>> Price's textbook "Infinitesimal Analsysis", >>>> from the mid 1700's through the late 1800's >>>> and fin-de-siecle, probably most closely match >>>> the fluxion and Leibniz's notions, our notions, >>>> while, "iota-values" are after the particular >>>> special character of the special function, >>>> the natural/unit equivalency function, in >>>> as with regards to plural: laws of large numbers, >>>> models of real numbers, definitions of continuity, >>>> models of Cantor space, and this as with being >>>> sets in a set theory, obviously extra-ordinary. >>>> >>>> Or, iota-values are consistent, and constructive, >>>> and their (relevant) properties decide-able, >>>> in descriptive set theories about a linear continuum, >>>> like today's most well-known, ZFC, and its models >>>> of a continuous domain: extent density completeness measure. >>>> >>>> >>> >>> >>> There's also Cavalieri to consider, >>> and Bradwardine from the Mertonian school >>> about De Continuo, where sometimes it's >>> said that Cavalieri in the time of Galileo >>> formalized infinitesimals. >>> >>> https://www.youtube.com/watch?v=EyWpZQny5cY&t=1590 >>> "Moment and Motion: meters, seconds, orders, inverses" >>> >>> Of course most people's usual ideas about >>> infinitesimals are what's called "atomism". >>> This is Democritus vis-a-vis Eudoxus. >>> >>> >>> Wow, it's like I just mentioned the conversation >>> here where was defined "continuous topology" >>> as "own initial and final topology". >>> >>> >> >> >> Cantor of course had an oft-repeated opinion >> on infinitesimals: "bacteria". This was after the >> current theory of the day of bacteria vis-a-vis miasma >> as the scientific source of disease, while these days >> it's known that there are symbiotic bacteria, >> while miasmas are still usually considered bad. >> He though was happy to ride Russell's retro-thesis, ========== REMAINDER OF ARTICLE TRUNCATED ==========