Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness)

On 9/24/2024 11:03 PM, Ross Finlayson wrote:

On 09/24/2024 02:47 PM, Ross Finlayson wrote:

On 09/24/2024 01:35 PM, Ross Finlayson wrote:

On 09/24/2024 10:22 AM, Jim Burns wrote:

On 9/20/2024 5:15 PM, Ross Finlayson wrote:

Then these lines-reals these iota-values
are about the only "standard infinitesimals"
there are: with extent you observe, density
you observe, least-upper-bound as trivial,
and measure as assigned, length assignment.

>
Lines with the least.upper.bound property
(equivalent to "crossing must intersect")
do not have infinitesimals.
>
For example,
there are no infinitesimals
between 0 and all the _finite_ unit.fractions.
>
⎛ Each positive point has
⎜ a finite unit.fraction between it and 0
⎜
⎜⎛ Otherwise,
⎜⎜ greatest.lower.bound β of finite unit.fractions
⎜⎜ is positive, and
⎜⎜ not.bounding 2⋅β > finite ⅟k
⎜⎜ ½⋅β > ¼⋅⅟k
⎜⎜ β > ½⋅β > ¼⋅⅟k
⎜⎜ greatest.lower.bound β is not.bounding,
⎝⎝ which is gibberish.

>
Well now, there are as many kinds infinitesimals
as there are infinities,

In this discussion, by 'infinitesimal', I mean
a point δ between 0 and all finite.unit.fractions.
infinitesimal δ :⇔
∀k ∈ ℕ:  0 < δ < ⅟k  ⇔
0 < δ ≤ᵉᵃᶜʰ ⅟ℕ
β = greatest.lower.bound of finite.unit.fractions
∀ᴿr: r ≤ᵉᵃᶜʰ ⅟ℕ  ⇒  r ≤ β ≤ᵉᵃᶜʰ ⅟ℕ
ℕ is well.ordered and (≠0) nexted
∀S ⊆ ℕ: S={}  ∨  ∃k ∈ S: k ≤ᵉᵃᶜʰ S
∀j ∈ ℕ: ∃!k ∈ ℕ\{0}: j+1=k
∀k ∈ ℕ\{0}: ∃!j ∈ ℕ: j+1=k
∀j ∈ ℕ: j < j+1  ∧  ¬∃k ∈ ℕ: j < k < j+1
lemma:
¬∃δ: 0 < δ ≤ᵉᵃᶜʰ ⅟ℕ
There are no infinitesimals like that.
My current purpose is to dissuade
believers in a smallest unit.fraction
from believing in a smallest unit.fraction.
Do you (RF) think that other systems of infinitesimals
might be of  use in that discussion?
If you think so, why do you?

and all in a general sense differing in
differences quite clustered about zero,
make for that Peano, Dodgson, Veronese,
Stolz, Leibniz, MacLaurin, Price,
the entire field of infinitesimal analysis as
what real analysis was named for hundreds of years,
make for that even Robinson's
rather modest and of no analytical character
the hyper-reals, or
as among Conway's surreal numbers,
has that most people's ideas of infinitesimals
are exactly as an infinite of them in [0,1],
constant monotone strictly increasing,
as with regards to "asymptotic equipartitioning"
and other aspects of higher, and lower, mathematics.
>
Newton's "fluxions", Aristotle's contemplations and
deliberations about atoms, Zeno's classical expositions,
quite a few of these have infinitesimals all quite
throughout every region of the linear continuum.
>
Maybe Hardy's pure mathematics makes for conflating
the objects of geometry, points and lines, with
a descriptive set theory's, a theory with only
one relation and only one-way, point-sets, yet
for making a theory with them all together,
makes for that since antiquity and through
today, notions like Bell's smooth analysis,
and Nelson's Internal Set Theory, if you
didn't know, each have that along the linear
continuum: are not "not infinitesimals".
>
Here these "iota-values" are considered
"standard infinitesimals".
>
Then, in the complete ordered field,
there's nothing to say about them
except nothing, well, some have that
its properties of least-upper-bound
and measure are actually courtesy already
a more fundamental continuum, in the theory,
as a constant, and not just stipulated
to match expectations.
>
The MacLaurin's infinitesimals and then for
Price's textbook "Infinitesimal Analsysis",
from the mid 1700's through the late 1800's
and fin-de-siecle, probably most closely match
the fluxion and Leibniz's notions, our notions,
while, "iota-values" are after the particular
special character of the special function,
the natural/unit equivalency function, in
as with regards to plural: laws of large numbers,
models of real numbers, definitions of continuity,
models of Cantor space, and this as with being
sets in a set theory, obviously extra-ordinary.
>
Or, iota-values are consistent, and constructive,
and their (relevant) properties decide-able,
in descriptive set theories about a linear continuum,
like today's most well-known, ZFC, and its models
of a continuous domain: extent density completeness measure.
>
>

>
>
There's also Cavalieri to consider,
and Bradwardine from the Mertonian school
about De Continuo, where sometimes it's
said that Cavalieri in the time of Galileo
formalized infinitesimals.
>
https://www.youtube.com/watch?v=EyWpZQny5cY&t=1590
"Moment and Motion: meters, seconds, orders, inverses"
>
Of course most people's usual ideas about
infinitesimals are what's called "atomism".
This is Democritus vis-a-vis Eudoxus.
>
>
Wow, it's like I just mentioned the conversation
here where was defined "continuous topology"
as "own initial and final topology".
>
>

  Cantor of course had an oft-repeated opinion
on infinitesimals: "bacteria". This was after the
current theory of the day of bacteria vis-a-vis miasma
as the scientific source of disease, while these days
it's known that there are symbiotic bacteria,
while miasmas are still usually considered bad.
He though was happy to ride Russell's retro-thesis,
after borrowing Heine's result in trigonometric series,
which though has some reasonings where it's not so,
and collecting the anti-diagonal from duBois-Reymond,
nested intervals from Pythagoras, and this kind of thing.
  Poincare didn't much concur.
 Euler'd been kind of like "notice I move things around
in my infinite series", with regards to it's sort of like
he took the maxim of the lever as that he was the origin.
Yet, the resulting Euler's identity and Eulerian/Gaussian
analysis has its own sort of crazing as with regards
to the veneer of the analysis. It's considered
"standard", though.
 Leibniz treated the differential as an algebraic
quantity, MacLaurin sort of righted that with
regards to fluent and fluxion, yet, sometimes
it's so, varies how and why it's so.
  Then, for something like George Berkley's
infinitesimals as "ghosts of a departed quantity",
you can read about as much into that as something
like George Carlin and "infinities are so profound I'm profane".
  Then after Leibniz there's nil-potent and particularly
nil-square, as what descends these days to that "the
only standard infinitesimal is zero", or as with regards
to that otherwise what results are tiny, yet standard
quantities, what in an infinite series may appreciate.
 Of course for things like non-linear analysis and where
what otherwise the nil-square washes out as obviated
by triangle, Cauchy-Schwartz, Holder inequality and so on,
it's part of the field about where it's resulted non-negligeable,
as with regards to usually nil-potent and nil-square.
  Anyways you can often find that if something like
Hilbert said that infinitary reasoning is the finest
creation of the human mind, it's in a wide variety.
  So, yeah, pretty much any matters of "non-" standard
analysis, of course in no way contradict standard real
analysis at all, instead their being "super-" standard.
 ... Because Eudoxus/Cauchy/Dedekind is insufficient, or,
at least it's known "at best: incomplete".
  Anyways: Democritus and atomism, and Eudoxus and the
Pythagorean and the Archimedean and the field, have
at least that Aristotle describes both as theoretically
so, like line-reals and field-reals, then though he
picks field-reals as he simply wasn't an atomist, for
Aristotle's substances and forms and Platonism.
 Which here is a retro-Heraclitan dual-monism so that
increment and equi-partition build arithmetic together.