Re: How many different unit fractions are lessorequal than all unit fractions? (iota-values)

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.
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.