On Thursday, May 26, 2022 at 12:59:14 PM UTC-7, Chris M. Thomasson wrote:
> On 5/26/2022 8:54 AM, Ross A. Finlayson wrote:
> > On Tuesday, May 24, 2022 at 10:26:01 PM UTC-7, zelos...@gmail.com wrote:
> >> tisdag 24 maj 2022 kl. 18:02:15 UTC+2 skrev Ross A. Finlayson:
> >>> On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
> >>>> tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> >>>>> On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> >>>>>> måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> >>>>>>> On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> >>>>>>>> torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> >>>>>>>>> and you get the first integer.
> >>>>>>>> There are no infinitesimals in real numbers.
> >>>>>>>>
> >>>>>>>> And 1=9/9=0.999...
> >>>>>>>>
> >>>>>>>> wrong as always
> >>>>>>> Are there infinite numbers in infinite numbers?
> >>>>>>>
> >>>>>>> If there are infinite numbers, they are infinite numbers
> >>>>>>> in infinite numbers.
> >>>>>> There are no "infinite numbers" in real numbers, real numbers are archimedian.
> >>>>>>>
> >>>>>>> Instead it's "for any large number, finite, there's
> >>>>>>> a larger one (also finite)" besides "for any large number,
> >>>>>>> finite, there's a large infinite, larger", from that
> >>>>>>> infinite numbers exist.
> >>>>>>>
> >>>>>>> This simply keeps what is quantitative there,
> >>>>>>> with respect to qualitative.
> >>>>> Also it is like an existence result itself,
> >>>>> that there are infinitely many
> >>>>> there are infintely grand.
> >>>>>
> >>>>> "Having the Archimedean property" is
> >>>>> often read two ways,
> >>>>> for the unbounded (not finitely many)
> >>>>> and the unbounded (not infinitely grand).
> >>>>>
> >>>>> It's kind of like Goedelian completeness:
> >>>>> reminding people of both the completeness
> >>>>> theorems, and the incompleteness theorems.
> >>>>>
> >>>>>
> >>>>> These days non-Archimedean fields are
> >>>>> a usual introduction to "non" standard ("extra" standard).
> >>>> They are not standard.
> >>>>>
> >>>>> Then, yes, I am talking about a logical consequence
> >>>>> of there being infinitely many that there are infinitesimals
> >>>>> in the reals and that besides there are infinites in integers.
> >>>> There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
> >>>>>
> >>>>> Then, what is the "standard" is just as above in matters of
> >>>>> "representation theory", here model theory for a function theory
> >>>>> for a space of values: it's standard and well-defined but not
> >>>>> complete, the space of representations those of the field reals,
> >>>>> Archimedean field reals, made replete with a space of
> >>>>> representations of those of line reals, or signal reals.
> >>>>>
> >>>>> That field reals, line reals, and signal reals, each in the
> >>>>> spaces of real values like usual vector spaces, are each
> >>>>> models of real numbers with IVT and resultingly the FTCs,
> >>>>> and otherwise real character: is central and important.
> >>>>>
> >>>>> (In mathematics.)
> >>>>>
> >>>>> It's kind of like "Burali-Forti's largest ordinal, that would
> >>>>> contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> >>>>> themselves contains itself": starting with that the only ordinals
> >>>>> are finite and Archimedean as you advise, that immediately any
> >>>>> "infinite" including omega or otherwise actual infinite:
> >>>>> includes itself. I.e. without "defining" omega all well-founded
> >>>>> and regular: it ("omega, an inductive set") would be "derived"
> >>>>> from the "paradoxes of Burali-Forti and Russell in an Archimedean
> >>>>> universe", as _not_ well-founded, regular, ordinary, ....
> >>>>>
> >>>>>
> >>>>> Then, for infinitesimals and the long line, which usually enough
> >>>>> abstractly includes infinitesimals, a usual enough notion of
> >>>>> the real line, partitions any segment into infinitely-many
> >>>>> equal-size pieces.
> >>>>>
> >>>>> Of course calculus was called "infinitesimal analysis" for
> >>>>> some hundreds of years, and that's what was meant, also.
> >>>>>
> >>>>> These days of course everybody knows Cauchy/Weierstrass as
> >>>>> the formalism after Riemann/Lebesgue the formalism, knowing
> >>>>> most all of a development of the complete ordered field (Archimedean),
> >>>>> besides usual graphical notions of the points that in their space
> >>>>> mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)
> >>>>>
> >>>>> Then, line reals, field reals, and signal reals are three _different_
> >>>>> models of real numbers, connected and having the gaplessness
> >>>>> property, least upper bound property, measure(s), ..., here of
> >>>>> course that line reals are modeled as "unboundedly many,
> >>>>> and, vanishingly small, and equal, values what sum to 1" .
> >>>>>
> >>>>>
> >>>>> Then where "the curriculum includes _only_ the field reals,
> >>>>> stop", it is short, because usual models of line reals and signal
> >>>>> reals besides field reals are everywhere and central, in all sorts
> >>>>> models in mathematics. So, the curriculum is short because
> >>>>> there are at least three _different_ models of "real numbers".
> >>> There are at least three models of reals, different sets,
> >>> and one of them is infinitesimals ("iota-values") zero to one.
> >>>
> >>> And, in their own way, the line reals are "Regular" and "Standard":
> >>> for example standardly modelling the Equivalency Function as a limit
> >>> of real functions.
> >>>
> >>> So, mathematics writ large is missing out from that the
> >>> properties of "real" infinites/infinitesimals: are direct and
> >>> consequent from the properties of the objects the numbers
> >>> their values themselves.
> >>>
> >>> If we can agree that "retro-finitism": is backward, then I hope
> >>> that you can see that ignoring these features, is also.
> >>>
> >>> I.e. for theories where these things exist, consistently of
> >>> course, not knowing them or ignoring them: is as bad as
> >>> retro-finitism.
> >>>
> >>> Which points all sorts arguments/rhetoric strongly back around - ....
> >> reals are by definition archimedian. There are no infinitesimals. If there are, they are not the real numbers.
> >
> > For foundations, we mostly look to "derive" instead of, "define", things.
> >
> > There isn't anywhere on the line for infinitesimals to be
> > that aren't already real numbers: that of course this
> > usual simple f(n) = n/d, n->d, d->oo, defines the extent,
> > bounds, density, LUB, and various measures (sigma algebras,
> > after length assignment), that result a model of reals
> > from what isn't not "infinitesimals".
> >
> > That fundamental theorems of calculus arrive from those
> > properties same for "field reals" and these "line reals", ...,
> > of course a formalist would demand. (If they are each
> > models, "sets of all reals", in an organization of set theory.)
> >
> > Really mostly about the continuous and discrete:
> > each from the other.
> >
> How small does a real number have to get in order for it to become
> "infinitesimal"? Or is that just a stupid question to begin with?
>
> I know this gets infinitesimal:
>
> [n] = .1^n
>
> And it's infinite set is comprised of real numbers. And yes, they can be
> plotted on a line.

An infinitesimal is what is smaller than 1/x for any positive, finite x.

Clearly, it's not 1/x for positive, finite x, because, 1/2x is 1/2 it, smaller.

So, the usual only infinitesimal is "zero", but actually, the very notion
of infinitesimal is "non-zero, smaller than the smallest increment,
is the smallest increment".

I.e. "smaller-than-itself" is about the nilpotent and nilsquare infinitesimals.
There are infinitesimals with properties as the nilpotent and nilsquare, just
like (or at least in metaphor) that there are infinites with properties as of
having various arthimetics, like (infinite) cardinals and ordinals.

Here the idea is that each 1/d has that {1+1+... d-many 1's} / d = 1.
As d goes to infinity, the idea is that the sum of the differences
that each make by dividing, what make these infinitesimals or
"iota-values", is always 1, and also that it's actually a sum or defined
on these iota-values pretty much exactly as if they were integers.

Of course pretty much everywhere when there's instructed calculus,
real analysis or calculus where of course there are any number of "calculi",
it's explained before Riemann sums that "these are not sums, and we have
a language of limit and the infinite limit so that infinite summability doesn't
possibly become non-sensible or non-sensical", not because "these sums
are not sums".

Then I suppose most people have infinitesimal as "width of a point".

Or, here, "side of a point that is in a line", or "sides of a point that is
on a line" - basically that an infinite is regular in [0,1], dense and, you know,
self-similar in all dimensions as it were: then those are infinitesimals,
besides something like iota-values with "iota-sums" and "iota-multiples",
about the nilpotent and nilsquare and so on, that these are infinitesimals
equipped in their space with arithmetic. (Iota-values.)

I heard of smooth infinitesimals before I heard of uncountable cardinals.

There are these iota-values between zero and one.

The differential and its notation "d": is most fundamental,
the derivative, what of course is the usual expression of a
real-valued variable like time, or length, dx/dy and the nilpotent
and nilsquare, differential, mostly reflect that "d" is read as
"an infinitesimal segment of".

Which of course is same as length assignment and iota-values.

Ahem.