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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? (iota-values)
Date: Mon, 23 Sep 2024 14:46:13 -0700
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On 9/23/2024 1:53 PM, Ross Finlayson wrote:
> On 09/22/2024 08:26 PM, Chris M. Thomasson wrote:
>> On 9/22/2024 7:47 PM, Ross Finlayson wrote:
>>> On 09/22/2024 02:12 PM, Chris M. Thomasson wrote:
>>>> On 9/22/2024 1:54 PM, Ross Finlayson wrote:
>>>>> On 09/22/2024 12:16 PM, Chris M. Thomasson wrote:
>>>>>> On 9/22/2024 12:09 PM, Ross Finlayson wrote:
>>>>>>> On 09/22/2024 11:54 AM, Chris M. Thomasson wrote:
>>>>>>>> On 9/22/2024 11:37 AM, WM wrote:
>>>>>>>>> On 22.09.2024 19:44, Jim Burns wrote:
>>>>>>>>>
>>>>>>>>>> There is no point next to 0.
>>>>>>>>>>
>>>>>>>>> This is definite: There is a smallest unit fraction because
>>>>>>>>> there are
>>>>>>>>> no unit fractions without a first one when counting from zero.
>>>>>>>>
>>>>>>>> Huh? Wow... Hummm... You suffer from some sort of learning
>>>>>>>> disorder? Or,
>>>>>>>> pure troll? Humm...
>>>>>>>>
>>>>>>>> There is no smallest unit fraction.
>>>>>>>
>>>>>>> In iota-values there is.
>>>>>>
>>>>>> The _smallest_ unit fraction, as in they are not infinite? Humm...
>>>>>> Keep
>>>>>> in mind that if you give me a unit fraction, I can always find a
>>>>>> smaller
>>>>>> one...
>>>>>>
>>>>>>
>>>>>>>
>>>>>>> That's what iota-values are, beyond the "infinite-divisible",
>>>>>>> the "infinitely-divided", _together_, as with regards to
>>>>>>> "asymptotic equipartitioning" and "uniformization in the limit",
>>>>>>> why it is so that what we were told in pre-calculus class,
>>>>>>> that 1/oo was not a thing, for the standard linear curriculum,
>>>>>>> has that it is a thing, and that this includes things like
>>>>>>> "I can interpret .999... as either ~1.0... or .997, .998, ...",
>>>>>>> with of course knowing when and where it's either way.
>>>>>>>
>>>>>>> Also this is one of Aristotle's notions, where Aristotle
>>>>>>> also more than 2000 years ago, describes "I can interpret .999..."
>>>>>>> about knowing which way is up.
>>>>>>>
>>>>>>> So, here sometimes it's called "Aristotle's continuum" as with
>>>>>>> regards to that otherwise of course the complete ordered field
>>>>>>> as Archimedes' and Eudoxus' continuum, later though Whig-ed out
>>>>>>> as it were with continental flavour, or Cauchy-Weierstrass, who
>>>>>>> give what's called "standard real analysis" these days.
>>>>>>>
>>>>>>> The idea of "iota" values as "standard infinitesimals"
>>>>>>> makes about most sense as that's what "iota" means, the word.
>>>>>>>
>>>>>>>
>>>>>>
>>>>>
>>>>> Nope, in iota-values, they're already smallest.
>>>>
>>>> What about an individual smallest unit fraction? You can say they get
>>>> arbitrarily close to zero, but that still does not mean there is a
>>>> smallest one...
>>>>
>>>>>
>>>>> If you look into "asymptotic equipartitioning" and
>>>>> for example "Jordan measure", in the "asymptotic equipartitioning"
>>>>> you can often find another "a.e.: almost everywhere",
>>>>> which is what happens when set theory results not being
>>>>> able to agree with itself, that purposefully and axiomatically
>>>>> it's stipulated to erase the difference, from "everywhere",
>>>>> which some see as an acceptable conceit, others as hypocritical,
>>>>> same thing.
>>>>>
>>>>> In field-reals of course there's that division is _closed_
>>>>> the operator, except of course usually division-by-zero,
>>>>> where of course delta-epsilonics builds a case for induction
>>>>> that "in the infinite limit" then that it goes to zero,
>>>>> "infinitesimal", in all the powers of division of integers.
>>>>>
>>>>> These though are "line-reals", another own "continuous domain",
>>>>> and constructively, also.
>>>>>
>>>>>
>>>>
>>>
>>> In field-reals there's no smallest magnitude non-zero,
>>> in line-reals there's a smallest magnitude non-zero.
>>>
>>> Each of field-reals and line-reals model a continuous
>>> domain, with the usual R, field-reals and then
>>> "a unit length interval, contiguous, defined for
>>> example as f(n) = n/d as 0 <= n <= d, d -> oo",
>>> then that a usual representation of any real
>>> magnitude or signed-magnitude is integer-part
>>> and non-integer part, the [0,1] being the non-integer
>>> part, and a dual representation bit for example just
>>> like field-reals have dual representation .999... = 1.
>>>
>>> Then, they don't go together and aren't added to together,
>>> field-reals and line-reals, except with regards to the
>>> book-keeping involved their values as magnitudes, and
>>> properties of those according to other matters of relation.
>>>
>>> I.e., they're not interchangeable, as they're not same,
>>> and have different definitions of continuity, yet in set
>>> theory they're each sets, then it results that just like
>>> there's a result in usual set theory that sets with
>>> different cardinals have no function between them,
>>> here it's that these are defined about their bounds
>>> and result being a "non-Cartesian", function in set
>>> theory what defines them, so, it's just another
>>> profound result in all of set theory.
>>>
>>> You might even figure it'll make the news someday.
>>>
>>> Here though it's old news.
>>
>> How does it define the smallest possible unit fraction? 1/n ? It cannot
>> be an actual example unit fraction because there will always be another
>> one that is smaller...
>>
> 
> Nope, it's defined for natural integers f(n) and there are
> no integers strictly between 0 and 1.

Oh. I failed to understand that you were talking about the granularity 
of integers. No integer between 0 and 1 for sure.

> 
> It's the properties of the entire range of this function
> what make it's fixed, scalarly, matters of perspective,
> and as with regards to that it's built before even
> considerations of otherwise fractions even exist.
> 
> Then, so defined as the range of a function, it's
> not a usual function, as it simply can't be re-ordered,
> as that's not how it's defined.
> 
> It's defined of very primitive elements and very simple
> aspects of laws of arithmetic, then with a usual apparatus
> of the infinite limit, et voila: iota-values, real-valued,
> according to laws of arithmetic.
> 
> It's not the field-reals, it's the line-reals, and it's
> shewn that as either is a continuous domain, either is a
> model of the Linear Continuum.
> 
> Then that one needs infinite limit and, the other needs
> not, yet the one without then must axiomatize itself
> its own completion according to Pythagoreans, lines-reals
> and field-reals, have here that Aristotle helps establish
> that since antiquity there are both considerations.
> 
>