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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: Sun, 22 Sep 2024 20:26:44 -0700
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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...