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

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:

On 09/20/2024 12:26 PM, Jim Burns wrote:

On 9/20/2024 2:10 PM, WM wrote:

On 20.09.2024 19:51, Jim Burns wrote:

>

Put pencil to paper and
 draw two curves which cross.
There is a point at which
 the curves intersect.

>

Theorems or axioms?

>
Here, a theorem.
⎛ (axiom)
⎜ The sets of ZFC exist.
⎜ (theorems)
⎜ ℕ exists
⎜ ℤ exists
⎜ ℚ exists
⎜ The set of Q.subsets
⎜ {S⊆ℚ:∅≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ\S≠∅}
⎜ exists and is the complete ordered field.
⎜ The Intermediate Value Theorem is true of
⎝ {S⊆ℚ:∅≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ\S≠∅}
>
Here, an axiom.
⎛ (axiom)
⎜ The IVT is true of ordered field 𝔽ᑉᐧⁱᵛᵗ
⎜ (theorem)
⎝ 𝔽ᑉᐧⁱᵛᵗ is Dedekind.complete, and thus is ℝ
>

"Drawing" a line, "tire en regle", or curve,
has that when you put pencil to paper,
and draw a line, or curve if you will,
and life the pencil and put it back down,
and draw another one, intersecting the first:
the _curves_ cross.
>
... At a point, of for example where
they're incident, they coincide.

>
Yes.
Because continuous curves must cross,
bounded nonempty set S must have a least.upper.bound.c
>
⎛ In particular, the function
⎜⎛ 0 above S
⎜⎝ 1 otherwise
⎜ doesn't intersect line y = 1/2  and
⎜ must be discontinuous somewhere  and
⎜ can only be discontinuous at lub.S  and
⎝ lub.S therefore exists.
>

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, 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".