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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 14:12:41 -0700
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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.
> 
>