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.