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

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:
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There is no point next to 0.
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This is definite: There is a smallest unit fraction because there are
no unit fractions without a first one when counting from zero.

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Huh? Wow... Hummm... You suffer from some sort of learning
disorder? Or,
pure troll? Humm...
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There is no smallest unit fraction.

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In iota-values there is.

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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...
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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.
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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.
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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.
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The idea of "iota" values as "standard infinitesimals"
makes about most sense as that's what "iota" means, the word.
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Nope, in iota-values, they're already smallest.

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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...
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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.
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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.
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These though are "line-reals", another own "continuous domain",
and constructively, also.
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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.
"real-valued"