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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
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Wed, 25 Sep 2024 11:44:22 -0700
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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,
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