Path: ...!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 05 Nov 2024 01:31:52 +0000 Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary) Newsgroups: sci.math References: <7def94cc-4a51-4305-8e62-0c5b5f5a6b0a@att.net> <2bb9c7a7-4fc9-497a-9f81-caea88dc0d4d@att.net> <01c88350-8726-4d1f-9c67-ec6dd699ef2d@att.net> <37d5162a-cd85-4ae8-a768-104746551ebb@att.net> <5fa0197c-6262-458a-b8e8-26237aadc118@att.net> From: Ross Finlayson Date: Mon, 4 Nov 2024 17:31:48 -0800 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: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 137 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-ZNyM1mSu0owvlLCX0u6AtwiDmmWhgysIQi9koXdGRil/rjuYVnDlSol5HwHNR0PGyNx7uCRj99y9E3K!ctNOVSj9pvoaXWRz7Bc3I6fm3kxxcPKgtdW8Dil9NGk4hLNBEuohdUUnaQKYIWr9QJrMoHFY2aU= 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: 6679 On 11/04/2024 03:09 PM, Jim Burns wrote: > On 11/4/2024 2:39 PM, Ross Finlayson wrote: >> On 11/04/2024 03:52 AM, Jim Burns wrote: >>> On 11/2/2024 6:01 PM, Ross Finlayson wrote: >>>> On 11/02/2024 12:37 PM, Jim Burns wrote: >>>>> On 11/2/2024 2:02 PM, Ross Finlayson wrote: > >>>>>> The delta-epsilonics of course, >>>>>> or some put it "delta-epsilontics", >>>>>> with little d and smaller e, >>>>>> of often for induction arbitrary m and larger n, >>>>>> is well-known to all students of calculus. >>>>>> "The infinitesimal analysis", .... >>>>> >>>>> The delta.epsilonics well.known to students of calculus >>>>> is not infinitesimal analysis. >>>>> For δ > 0 and ε > 0 >>>>> there are _finite_ j and k such that >>>>> δ > ⅟j > 0 and ε > ⅟k > 0 > >> Sure it is, >> the delta-epsilonics is well known, > > For δ > 0 > there is finite j such that δ > ⅟j > 0 > > ⎛ Assume otherwise. > ⎜ Assume, for δ > 0, that > ⎜ no ⅟j exists: δ > ⅟j > 0 > ⎜ δ is a lower bound of ⅟ℕ > ⎜ ⅟ℕ = {⅟i: i ∈ ℕ⁺ ∧ i finite} > ⎜ > ⎜ Let β be the greatest lower bound of ⅟ℕ > ⎜ β ≥ δ > 0 > ⎜ 2⋅β > β > ½⋅β > ⎜ > ⎜ ⅟ℕ ᵉᵃᶜʰ≥ β > ½⋅β > ⎜ ⅟ℕ ᵉᵃᶜʰ≥ ½⋅β > ⎜ ½⋅β is a lower bound of ⅟ℕ [1] > ⎜ > ⎜ If 2⋅β is a lower.bound of ⅟ℕ > ⎜ then 2.β > β is a greater.than.β lower bound of ⅟ℕ > ⎜ which is not a thing. > ⎜ 2⋅β is not a lower bound of ⅟ℕ > ⎜ > ⎜ 2⋅β is not a lower bound of ⅟ℕ > ⎜ exists ⅟k ∈ ⅟ℕ: ⅟k > 2⋅β > ⎜ exists ¼⋅⅟k ∈ ⅟ℕ: ¼⋅⅟k > ½⋅β > ⎜ ½⋅β is not a lower bound of ⅟v ℕ > ⎜ > ⎜ However, > ⎜ [1] ½⋅β is a lower bound of ⅟ℕ > ⎝ Contradiction. > > Therefore, > For δ > 0 > there is finite j such that δ > ⅟j > 0 > >> What I'm saying is that since antiquity, >> it is known, >> that there are at least two models of continuity, >> and you may call it Archimedean and Democritan, >> about the field of rationals versus atomism, >> and that infinitesimal analysis includes both. > > Calculus class. > Complete ordered field. > Delta.epsilonics. > No infinitesimals. > >> So, infinitesimal analysis includes delta-epsilonics, >> if not the other way around. > > Infinitesimal analysis without infinitesimals. > So creative! > > > Actually, in infinitesimal analysis, there's a reasoning that there _are_ infinitesimals in the geometric series, that the limit, to exist, _is_ the sum, so, in infinitesimal analysis, it's merely a simple definition of limit that thusly "attains to" the limit if never reaching it, that in infinitesimal analysis, "reaches". In that sense it's merely a polite conceit. That is to say, and I hope it's clear when mentioned, there's a reasoning that the limit must be the infinite limit and must actually as a matter of "fact" _reach_ the limit to exist - because, for example, then in the integral calculus, "the sum of sums" must make for not letting a case for "sum of zeros". I.e., it's certainly agreeable dealing with any derived quantity that it "is" itself, ..., less so that it isn't. Then of course if not one would needfully declare that all the declarations are cumulative and the notation representing a manner-of-speaking, this "integral" and "differential", about "infinity" and, ..., "zero", it's certainly so that some have the infinite limit _is_ the sum, and not merely "no different". So, in infinitesimal analysis, it's a sort of constructive development, delta-epsilonics and the limit, that is to say that constructivist means avoiding proof-by-contradiction, with regards to "there is ..." and "there isn't not ...". It's a sort of not constructivist development, while it's agreeably constructive. I.e., in infinitesimal analysis or "where quantities, even non-zero infinitesimals exist", then it's the same sort of development, yet for finitists it never completes while for the infinitary it's perfect. Then, some maintain that to be correct it's perfect, and that the limit _is_ the sum, because, otherwise there's a reductio where it's also _less_ the sum. Yet, people are told it's acceptable to make do, and told that it suffices. Others simply have it not so. Hey thanks though, it's spirited the craigh, yet you'll see again it's not quite entailed from the wider milieu.