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

On 11/04/2024 03:52 AM, Jim Burns wrote:

On 11/2/2024 6:01 PM, Ross Finlayson wrote:

On 11/02/2024 12:37 PM, Jim Burns wrote:

On 11/2/2024 2:02 PM, Ross Finlayson wrote:

>

The delta-epsilonics of course,
or some put it "delta-epsilontics",
with little d and smaller e,

>

of often for induction arbitrary m and larger n,
is well-known to all students of calculus.
>
"The infinitesimal analysis", ....

>
The delta.epsilonics well.known to students of calculus
is not infinitesimal analysis.
For δ > 0 and ε > 0
there are _finite_ j and k such that
δ > ⅟j > 0 and ε > ⅟k > 0

>

The delta-epsilonics is
a perfectly suitable approach to defining
infinite limit in theories of infinitesimal analysis.

>
The delta.epsilonics well.known to students of calculus
is not infinitesimal analysis.
For δ > 0 and ε > 0
there are _finite_ j and k such that
δ > ⅟j > 0 and ε > ⅟k > 0
>

How about atomism, is there a theory there with
truly abstractly uncuttable objects like the
theoretical atom?

>
There are lots and lots of theories.
>
There is one theory
(modulo lots and lots of ways to express it)
of the rationals with just enough points more that
continuous curves which cross must intersect.
>
That one theory is
the theory well.known to students of calculus.
>
I mention this
(and mention it and mention it)
because
I am declining your invitation to
be the mark in a bait.and.switch.
>
>

Sure it is, the delta-epsilonics is well known,
and where often proofs in geometry will have been
introduced vis-a-vis usual sorts of plain "checks"
as "proofs", then delta-epsilonics is often the
first sort of deductive account, introduced to
what's usually upper-class secondary students,
then that the concept of the infinite limit
is made and then it's pointed out how that fits
with regards to classical expositions like Zeno's,
of what yet doesn't.
What I'm saying is that since antiquity,
it is known,
that there are at least two models of continuity,
and you may call it Archimedean and Democritan,
about the field of rationals versus atomism,
and that infinitesimal analysis includes both.
So, no, I'm not interested nor was it proffered
"bait-and-switch", though here there's still
the "pick one, get both".
The "clock arithmetic" is usually fmailiar to
students as with regards to the sweep of
the secondae hand around the clock, or as
with regards to the rollover of the odometer,
that a course-of-passage in _time_ is most
simply as in accords with the classical expositions
of constant and uniform motion.
Indeed the dialectic I should hope you know as
something akin to "thesis, anti-thesis, synthesis",
as with regards to that being "pick one, get both".
I.e., the complementary duals, these competing concerns,
meet in the middle, even if: "the middle of nowhere".
So, infinitesimal analysis includes delta-epsilonics,
if not the other way around.
Then, some "continuum infinitesimal analysis",
makes for "Standard Infinitesimals" like
these "iota-values" of "line-reals".
constant monotone strictly increasing
extent density completeness measure