On 5/24/2022 6:50 PM, Chris M. Thomasson wrote:
> On 5/24/2022 4:37 PM, FromTheRafters wrote:
>> Chris M. Thomasson wrote :
>>> On 5/24/2022 1:43 AM, Reef Kubo wrote:
>>>> Chris M. Thomasson wrote:
>>>>
>>>>>> and you get the first integer.
>>>>>
>>>>> Is your infinitesimal something like: .1^(infinity) ?
>>>>> .1^5 = .00001 .1^6 = .000001 ...
>>>>> It gets pretty damn small! Run for infinity the limit is zero, right? So
>>>>> if zero is the limit of the infinitesimal .000...1, and one is the limit
>>>>> of .999..., then that just means: 1 + 0 = 1
>>>>
>>>> idiot, there is no "damn small" in mathematics. If your region of interest
>>>> is bellow that limit, that number can be huge, not small.
>>>
>>> Well, this does get infinitely small with a step-by-step process:
>>>
>>> .1^1 = .1
>>> .1^2 = .01
>>> .1^3 = .001
>>> .1^4 = .0001
>>> .1^5 = .00001
>>> .1^6 = .000001
>>> ...
>>>
>>> Agreed?
>>
>> It gets smaller with each of infinitely many steps.
>>
>> What's the series (sum)?
>
> Ahh, the sum... Well, what about this:
>
> .1 + .01 = .11
>
>
> Okay? What about:
>
> .11 + .001 = .111
>
> Humm... There is a pattern. .111... forever?
>
> .111 + .0001 = .1111
>
> Ahhh... .1111 * 9 = .9999 so getting bigger, yet confined within a limit.

should add some of the preceding and following 0's for completeness

....0000000000.11110000000000000... * ...0000009.0000000... =

....0000000000.99990000000000000.....

I think all those 000s really just wrap around from back to front again, in a huge loop around the back of the page

EZ er to read version;

....0,000,000,000.99990000000000000

On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> > Just because in theory an infinite number of steps is required doesn't
> > mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
> > races a tortoise with a head start. Each time Achilles reaches a point
> > where the tortoise was, the tortoise advances somewhat. When Achilles
> > reaches that point, the tortoise advances more. And so on for an
> > infinite number of steps. Yet Achilles catches up to the tortoise and
> > passes it and wins the race, despite taking an infinite number of steps
> > to catch up to the tortoise.
> If Achilles strictly plays by the tortoises rules on a step-by-step basis:
>
> step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
> is ahead because of the head start.
>
> step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
> is still ahead because of the head start.
>
> on and on. The turtle will cross the finish line before Achilles.

The turtle will never cross the finish line but will always be ahead of Achilles.

> Now, if Achilles tells the tortoise to f-off and just starts running, he
> will quickly pass the tortoise...
>
> ;^)

On 5/25/2022 9:21 PM, Dan joyce wrote:
> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>> Just because in theory an infinite number of steps is required doesn't
>>> mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
>>> races a tortoise with a head start. Each time Achilles reaches a point
>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>> reaches that point, the tortoise advances more. And so on for an
>>> infinite number of steps. Yet Achilles catches up to the tortoise and
>>> passes it and wins the race, despite taking an infinite number of steps
>>> to catch up to the tortoise.
>> If Achilles strictly plays by the tortoises rules on a step-by-step basis:
>>
>> step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
>> is ahead because of the head start.
>>
>> step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
>> is still ahead because of the head start.
>>
>> on and on. The turtle will cross the finish line before Achilles.
>
> The turtle will never cross the finish line but will always be ahead of Achilles.

both turtle and Achilles are points, they have no physical bodies, as this is a thought problem.

>
>> Now, if Achilles tells the tortoise to f-off and just starts running, he
>> will quickly pass the tortoise...
>>
>> ;^)

Using his foot, Achilles could tip the turtle from the rear onto his back, little turtle legs a flailing in the air...

On 5/25/2022 7:21 PM, Dan joyce wrote:
> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>> Just because in theory an infinite number of steps is required doesn't
>>> mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
>>> races a tortoise with a head start. Each time Achilles reaches a point
>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>> reaches that point, the tortoise advances more. And so on for an
>>> infinite number of steps. Yet Achilles catches up to the tortoise and
>>> passes it and wins the race, despite taking an infinite number of steps
>>> to catch up to the tortoise.
>> If Achilles strictly plays by the tortoises rules on a step-by-step basis:
>>
>> step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
>> is ahead because of the head start.
>>
>> step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
>> is still ahead because of the head start.
>>
>> on and on. The turtle will cross the finish line before Achilles.
>
> The turtle will never cross the finish line but will always be ahead of Achilles.

Yes. True. It gets infinitely closer and closer to the finish line.

>
>> Now, if Achilles tells the tortoise to f-off and just starts running, he
>> will quickly pass the tortoise...
>>
>> ;^)

On Sat May 21 10:38:35 2022 "mitchr...@gmail.com" wrote:
> On Thursday, May 19, 2022 at 2:12:58 PM UTC-7, Michael Moroney wrote:
> > Of course. 1+0=1.
>
> But .999... plus zero equals .999...
> To get to 1 you need to add an infinitesimal instead.
>
> Mitchell Raemsch

*
Mitchell: What is the definition of the "infintesimal"?

Does it lie on the number line?

earle
*

On 5/25/2022 9:27 PM, Earle Jones wrote:
> On Sat May 21 10:38:35 2022 "mitchr...@gmail.com" wrote:
>> On Thursday, May 19, 2022 at 2:12:58 PM UTC-7, Michael Moroney wrote:
>>> Of course. 1+0=1.
>>
>> But .999... plus zero equals .999...
>> To get to 1 you need to add an infinitesimal instead.
>>
>> Mitchell Raemsch
>
> *
> Mitchell: What is the definition of the "infintesimal"?
>
> Does it lie on the number line?
>
> earle
> *

The only thing I can think of wrt one of his "infinitesimal's" is the
following:

..1^0 = 1
..1^1 = .1
..1^2 = .01
..1^3 = .001
..1^4 = .0001
..1^5 = .00001
..1^6 = .000001
....

[n] = .1^n

where n is an unsigned integer.

It gets smaller and smaller. It's limit is zero.

On Wednesday, May 25, 2022 at 9:27:45 PM UTC-7, Earle Jones wrote:
> On Sat May 21 10:38:35 2022 "mitchr...@gmail.com" wrote:
> > On Thursday, May 19, 2022 at 2:12:58 PM UTC-7, Michael Moroney wrote:
> > > Of course. 1+0=1.
> >
> > But .999... plus zero equals .999...
> > To get to 1 you need to add an infinitesimal instead.
> >
> > Mitchell Raemsch
>
> *
> Mitchell: What is the definition of the "infintesimal"?
>
> Does it lie on the number line?
>
> earle
> *

One divided by the infinite.
You can plot it as the first quantity after zero.

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.

Ross A. Finlayson wrote:

>> > 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.

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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.

On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> On 5/25/2022 7:21 PM, Dan joyce wrote:
>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>>> Just because in theory an infinite number of steps is required doesn't
>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
>>>> Achilles
>>>> races a tortoise with a head start. Each time Achilles reaches a point
>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>>> reaches that point, the tortoise advances more.  And so on for an
>>>> infinite number of steps.  Yet Achilles catches up to the tortoise and
>>>> passes it and wins the race, despite taking an infinite number of steps
>>>> to catch up to the tortoise.
>>> If Achilles strictly plays by the tortoises rules on a step-by-step
>>> basis:

What are "the tortoise's rules"? The only rules are the tortoise gets a
head start and both it and Achilles run as fast as they can to the
finish line, and whoever does so first, wins.
>>>
>>> step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
>>> is ahead because of the head start.
>>>
>>> step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
>>> is still ahead because of the head start.
>>>
>>> on and on. The turtle will cross the finish line before Achilles.
>>
>> The turtle will never cross the finish line but will always be ahead
>> of Achilles.
>
> Yes. True. It gets infinitely closer and closer to the finish line.

That is not Zeno's Paradox. The tortoise gets a head start, at point
A[1]. The race starts. When Achilles reaches A[1], the tortoise has
moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so forth.

Since Achilles is faster than the tortoise, the distances A[1], A[2],
[A3], ... get smaller and smaller, since the time it takes Achilles to
run from the start to A[1] equals the time it takes the slower tortoise
to run from A[1] to A[2], and so on.

The paradox is, no matter how big n gets, A[n] (Achilles' position) is
always behind A[n+1] (the tortoise's position), even as n approaches
infinity. So Achilles can never beat the tortoise, right? But, as long
as the head start isn't _too_ large, in real life, Achilles passes the
tortoise and wins, just as you'd expect. So what's wrong with this?

As I said, just because there's an infinite limit, it doesn't mean the
limit is absolute. In this case, the total time passed also reaches a
limit (at n=infinity) but that time limit isn't infinite, so what
happens after the "limit" on time passes? As always, time marches on...
At that point Achilles passes the tortoise and remains ahead for the
rest of the race, and the infinite series no longer applies.
>
>
>>> Now, if Achilles tells the tortoise to f-off and just starts running, he
>>> will quickly pass the tortoise...

In real life, yes, but in Zeno's Paradox, no.
>>>
>>> ;^)
>

Extra credit: Given the speeds of Achilles S1 and the tortoise S2
(S1>S2), and the head start distance A1, how long does it take for
Achilles to pass the tortoise? :-)

On 5/26/2022 1:25 PM, Michael Moroney wrote:
> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
>> On 5/25/2022 7:21 PM, Dan joyce wrote:
>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>>>> Just because in theory an infinite number of steps is required doesn't
>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
>>>>> Achilles
>>>>> races a tortoise with a head start. Each time Achilles reaches a point
>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>>>> reaches that point, the tortoise advances more.  And so on for an
>>>>> infinite number of steps.  Yet Achilles catches up to the tortoise and
>>>>> passes it and wins the race, despite taking an infinite number of
>>>>> steps
>>>>> to catch up to the tortoise.
>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
>>>> basis:
>
> What are "the tortoise's rules"?  The only rules are the tortoise gets a
> head start and both it and Achilles run as fast as they can to the
> finish line, and whoever does so first, wins.
>>>>
>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
>>>> tortoise
>>>> is ahead because of the head start.
>>>>
>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
>>>> tortoise
>>>> is still ahead because of the head start.
>>>>
>>>> on and on. The turtle will cross the finish line before Achilles.
>>>
>>> The turtle will never cross the finish line but will always be ahead
>>> of Achilles.
>>
>> Yes. True. It gets infinitely closer and closer to the finish line.
>
> That is not Zeno's Paradox.  The tortoise gets a head start, at point
> A[1].  The race starts.  When Achilles reaches A[1], the tortoise has
> moved ahead somewhat, to what we call A[2].  When Achilles reaches A[2].
> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> forth.
>
> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> [A3], ... get smaller and smaller, since the time it takes Achilles to
> run from the start to A[1] equals the time it takes the slower tortoise
> to run from A[1] to A[2], and so on.
>
> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> always behind A[n+1] (the tortoise's position), even as n approaches
> infinity. So Achilles can never beat the tortoise, right?  But, as long
> as the head start isn't _too_ large, in real life, Achilles passes the
> tortoise and wins, just as you'd expect. So what's wrong with this?
>
> As I said, just because there's an infinite limit, it doesn't mean the
> limit is absolute. In this case, the total time passed also reaches a
> limit (at n=infinity) but that time limit isn't infinite, so what
> happens after the "limit" on time passes? As always, time marches on...
> At that point Achilles passes the tortoise and remains ahead for the
> rest of the race, and the infinite series no longer applies.
>>
>>
>>>> Now, if Achilles tells the tortoise to f-off and just starts
>>>> running, he
>>>> will quickly pass the tortoise...
>
> In real life, yes, but in Zeno's Paradox, no.
>>>>
>>>> ;^)
>>
>
> Extra credit:  Given the speeds of Achilles S1 and the tortoise S2
> (S1>S2), and the head start distance A1, how long does it take for
> Achilles to pass the tortoise?  :-)
>

I did some equations on this a while back:

https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ

Here are my comments:

Iirc, scale was speed:
____________________________
[...]
Ahhhh, now this is a direct formula:

n = iteration count
d = distance
s = scale

r_[n] = (d / s^n) * (s^n - (s-1)^n)

just might work for finding the total distance
traveled at a given iteration count of the following
iterated equation:

r_[n+1] = r_[n] + (d - r_[n]) / s

Here is the sequence for d = 10 and s = 4 using the
iterative formula:
__________________________________
r_[0] = 0
r_[1] = 0 + (10 - 0) / 4 = 2.5
r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
__________________________________

And here is the sequence for d = 10 and s = 4 using
the direct formula:
__________________________________
r_[0] = 10 / 1 * 0 = 0
r_[1] = 10 / 4 * 1 = 2.5
r_[2] = 10 / 16 * 7 = 4.375
r_[3] = 10 / 64 * 37 = 5.78125
r_[4] = 10 / 256 * 175 = 6.8359375
__________________________________

As you can see, they are identical!

Humm...
____________________________

Here is another post:

https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
____________________________
I think I found a way to find the handicap of a
runner in an infinite race on a finite track...

How about something like:

Let:

d = total distance in track
s = scale, which relates to speed
n = integer iteration count, which relates to time
r_h = a runners starting handicap

Here is the iterative equation for finding the
distance a runner is down the track that I posted
up thread:

r_[n + 1] = r_[n] + (d - r_[n]) / s

The handicap of the runner is equal to r_[0]
because n = 0 is the starting position of every
runner.

The goal is to find the handicap of a runner with
a given distance, iteration count, total distance
of the track, and a scale or speed. AFAICT, the
following formula solves for the handicap of a
runner using that information:

r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)

Here is output of a racer using the iterative equation
with the following attributes:

d = 10
s = 4
r_h = 6.8
_______________________________________
r_[0] = 6.8
r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
_______________________________________

As we can see this runner has a head start of 6.8 out
of 10. Also, in the third frame, the runner r_[2] has
traveled 8.2 out of a possible 10.0.

Given that information alone, we can plug it all into
the formula for finding the handicap, and get:

r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8

Bingo! We now know that the handicap for the runner
is 6.8 at n = 0 by information reaped in a later moment
in time when n = 2... Three frames later.

Is this Kosher?!?!

:^o

____________________________

On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> > On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> >> On 5/25/2022 7:21 PM, Dan joyce wrote:
> >>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> >>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> >>>>> Just because in theory an infinite number of steps is required doesn't
> >>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> >>>>> Achilles
> >>>>> races a tortoise with a head start. Each time Achilles reaches a point
> >>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> >>>>> reaches that point, the tortoise advances more. And so on for an
> >>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> >>>>> passes it and wins the race, despite taking an infinite number of
> >>>>> steps
> >>>>> to catch up to the tortoise.
> >>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> >>>> basis:
> >
> > What are "the tortoise's rules"? The only rules are the tortoise gets a
> > head start and both it and Achilles run as fast as they can to the
> > finish line, and whoever does so first, wins.
> >>>>
> >>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> >>>> tortoise
> >>>> is ahead because of the head start.
> >>>>
> >>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> >>>> tortoise
> >>>> is still ahead because of the head start.
> >>>>
> >>>> on and on. The turtle will cross the finish line before Achilles.
> >>>
> >>> The turtle will never cross the finish line but will always be ahead
> >>> of Achilles.
> >>
> >> Yes. True. It gets infinitely closer and closer to the finish line.
> >
> > That is not Zeno's Paradox. The tortoise gets a head start, at point
> > A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> > moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> > the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> > forth.
> >
> > Since Achilles is faster than the tortoise, the distances A[1], A[2],
> > [A3], ... get smaller and smaller, since the time it takes Achilles to
> > run from the start to A[1] equals the time it takes the slower tortoise
> > to run from A[1] to A[2], and so on.
> >
> > The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> > always behind A[n+1] (the tortoise's position), even as n approaches
> > infinity. So Achilles can never beat the tortoise, right? But, as long
> > as the head start isn't _too_ large, in real life, Achilles passes the
> > tortoise and wins, just as you'd expect. So what's wrong with this?
> >
> > As I said, just because there's an infinite limit, it doesn't mean the
> > limit is absolute. In this case, the total time passed also reaches a
> > limit (at n=infinity) but that time limit isn't infinite, so what
> > happens after the "limit" on time passes? As always, time marches on...
> > At that point Achilles passes the tortoise and remains ahead for the
> > rest of the race, and the infinite series no longer applies.
> >>
> >>
> >>>> Now, if Achilles tells the tortoise to f-off and just starts
> >>>> running, he
> >>>> will quickly pass the tortoise...
> >
> > In real life, yes, but in Zeno's Paradox, no.
> >>>>
> >>>> ;^)
> >>
> >
> > Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> > (S1>S2), and the head start distance A1, how long does it take for
> > Achilles to pass the tortoise? :-)
> >
> I did some equations on this a while back:
>
> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
>
> Here are my comments:
>
> Iirc, scale was speed:
> ____________________________
> [...]
> Ahhhh, now this is a direct formula:
>
> n = iteration count
> d = distance
> s = scale
>
> r_[n] = (d / s^n) * (s^n - (s-1)^n)
>
>
> just might work for finding the total distance
> traveled at a given iteration count of the following
> iterated equation:
>
> r_[n+1] = r_[n] + (d - r_[n]) / s
>
>
>
> Here is the sequence for d = 10 and s = 4 using the
> iterative formula:
> __________________________________
> r_[0] = 0
> r_[1] = 0 + (10 - 0) / 4 = 2.5
> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> __________________________________
>
>
> And here is the sequence for d = 10 and s = 4 using
> the direct formula:
> __________________________________
> r_[0] = 10 / 1 * 0 = 0
> r_[1] = 10 / 4 * 1 = 2.5
> r_[2] = 10 / 16 * 7 = 4.375
> r_[3] = 10 / 64 * 37 = 5.78125
> r_[4] = 10 / 256 * 175 = 6.8359375
> __________________________________
>
>
> As you can see, they are identical!
>
> Humm...
> ____________________________
>
>
> Here is another post:
>
> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> ____________________________
> I think I found a way to find the handicap of a
> runner in an infinite race on a finite track...
>
> How about something like:
>
>
> Let:
>
> d = total distance in track
> s = scale, which relates to speed
> n = integer iteration count, which relates to time
> r_h = a runners starting handicap
>
>
>
> Here is the iterative equation for finding the
> distance a runner is down the track that I posted
> up thread:
>
> r_[n + 1] = r_[n] + (d - r_[n]) / s
>
>
> The handicap of the runner is equal to r_[0]
> because n = 0 is the starting position of every
> runner.
>
> The goal is to find the handicap of a runner with
> a given distance, iteration count, total distance
> of the track, and a scale or speed. AFAICT, the
> following formula solves for the handicap of a
> runner using that information:
>
>
> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
>
>
>
> Here is output of a racer using the iterative equation
> with the following attributes:
>
> d = 10
> s = 4
> r_h = 6.8
> _______________________________________
> r_[0] = 6.8
> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> _______________________________________
>
>
>
> As we can see this runner has a head start of 6.8 out
> of 10. Also, in the third frame, the runner r_[2] has
> traveled 8.2 out of a possible 10.0.
>
> Given that information alone, we can plug it all into
> the formula for finding the handicap, and get:
>
>
> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
>
>
>
> Bingo! We now know that the handicap for the runner
> is 6.8 at n = 0 by information reaped in a later moment
> in time when n = 2... Three frames later.
>
>
> Is this Kosher?!?!
>
>
>
> :^o
>
> ____________________________

If you add zero to .999 repeating you still get .999 repeating.
Add the infinitely small and you get 1 instead.

On Thu May 26 04:27:35 2022 Earle Jones wrote:
> On Sat May 21 10:38:35 2022 "mitchr...@gmail.com" wrote:
> > On Thursday, May 19, 2022 at 2:12:58 PM UTC-7, Michael Moroney wrote:
> > > Of course. 1+0=1.
> >
> > But .999... plus zero equals .999...
> > To get to 1 you need to add an infinitesimal instead.
> >
> > Mitchell Raemsch
>
> *
> Mitchell: What is the definition of the "infintesimal"?
>
> Does it lie on the number line?
>
> earle
> *

*
Another question for Mitchell:

What is 1/INF ? (INF = the infintesimal)

Thanks,

earle
*

On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
>> On 5/26/2022 1:25 PM, Michael Moroney wrote:
>>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
>>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
>>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>>>>>> Just because in theory an infinite number of steps is required doesn't
>>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
>>>>>>> Achilles
>>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
>>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>>>>>> reaches that point, the tortoise advances more. And so on for an
>>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
>>>>>>> passes it and wins the race, despite taking an infinite number of
>>>>>>> steps
>>>>>>> to catch up to the tortoise.
>>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
>>>>>> basis:
>>>
>>> What are "the tortoise's rules"? The only rules are the tortoise gets a
>>> head start and both it and Achilles run as fast as they can to the
>>> finish line, and whoever does so first, wins.
>>>>>>
>>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
>>>>>> tortoise
>>>>>> is ahead because of the head start.
>>>>>>
>>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
>>>>>> tortoise
>>>>>> is still ahead because of the head start.
>>>>>>
>>>>>> on and on. The turtle will cross the finish line before Achilles.
>>>>>
>>>>> The turtle will never cross the finish line but will always be ahead
>>>>> of Achilles.
>>>>
>>>> Yes. True. It gets infinitely closer and closer to the finish line.
>>>
>>> That is not Zeno's Paradox. The tortoise gets a head start, at point
>>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
>>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
>>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
>>> forth.
>>>
>>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
>>> [A3], ... get smaller and smaller, since the time it takes Achilles to
>>> run from the start to A[1] equals the time it takes the slower tortoise
>>> to run from A[1] to A[2], and so on.
>>>
>>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
>>> always behind A[n+1] (the tortoise's position), even as n approaches
>>> infinity. So Achilles can never beat the tortoise, right? But, as long
>>> as the head start isn't _too_ large, in real life, Achilles passes the
>>> tortoise and wins, just as you'd expect. So what's wrong with this?
>>>
>>> As I said, just because there's an infinite limit, it doesn't mean the
>>> limit is absolute. In this case, the total time passed also reaches a
>>> limit (at n=infinity) but that time limit isn't infinite, so what
>>> happens after the "limit" on time passes? As always, time marches on...
>>> At that point Achilles passes the tortoise and remains ahead for the
>>> rest of the race, and the infinite series no longer applies.
>>>>
>>>>
>>>>>> Now, if Achilles tells the tortoise to f-off and just starts
>>>>>> running, he
>>>>>> will quickly pass the tortoise...
>>>
>>> In real life, yes, but in Zeno's Paradox, no.
>>>>>>
>>>>>> ;^)
>>>>
>>>
>>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
>>> (S1>S2), and the head start distance A1, how long does it take for
>>> Achilles to pass the tortoise? :-)
>>>
>> I did some equations on this a while back:
>>
>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
>>
>> Here are my comments:
>>
>> Iirc, scale was speed:
>> ____________________________
>> [...]
>> Ahhhh, now this is a direct formula:
>>
>> n = iteration count
>> d = distance
>> s = scale
>>
>> r_[n] = (d / s^n) * (s^n - (s-1)^n)
>>
>>
>> just might work for finding the total distance
>> traveled at a given iteration count of the following
>> iterated equation:
>>
>> r_[n+1] = r_[n] + (d - r_[n]) / s
>>
>>
>>
>> Here is the sequence for d = 10 and s = 4 using the
>> iterative formula:
>> __________________________________
>> r_[0] = 0
>> r_[1] = 0 + (10 - 0) / 4 = 2.5
>> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
>> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
>> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
>> __________________________________
>>
>>
>> And here is the sequence for d = 10 and s = 4 using
>> the direct formula:
>> __________________________________
>> r_[0] = 10 / 1 * 0 = 0
>> r_[1] = 10 / 4 * 1 = 2.5
>> r_[2] = 10 / 16 * 7 = 4.375
>> r_[3] = 10 / 64 * 37 = 5.78125
>> r_[4] = 10 / 256 * 175 = 6.8359375
>> __________________________________
>>
>>
>> As you can see, they are identical!
>>
>> Humm...
>> ____________________________
>>
>>
>> Here is another post:
>>
>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
>> ____________________________
>> I think I found a way to find the handicap of a
>> runner in an infinite race on a finite track...
>>
>> How about something like:
>>
>>
>> Let:
>>
>> d = total distance in track
>> s = scale, which relates to speed
>> n = integer iteration count, which relates to time
>> r_h = a runners starting handicap
>>
>>
>>
>> Here is the iterative equation for finding the
>> distance a runner is down the track that I posted
>> up thread:
>>
>> r_[n + 1] = r_[n] + (d - r_[n]) / s
>>
>>
>> The handicap of the runner is equal to r_[0]
>> because n = 0 is the starting position of every
>> runner.
>>
>> The goal is to find the handicap of a runner with
>> a given distance, iteration count, total distance
>> of the track, and a scale or speed. AFAICT, the
>> following formula solves for the handicap of a
>> runner using that information:
>>
>>
>> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
>>
>>
>>
>> Here is output of a racer using the iterative equation
>> with the following attributes:
>>
>> d = 10
>> s = 4
>> r_h = 6.8
>> _______________________________________
>> r_[0] = 6.8
>> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
>> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
>> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
>> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
>> _______________________________________
>>
>>
>>
>> As we can see this runner has a head start of 6.8 out
>> of 10. Also, in the third frame, the runner r_[2] has
>> traveled 8.2 out of a possible 10.0.
>>
>> Given that information alone, we can plug it all into
>> the formula for finding the handicap, and get:
>>
>>
>> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
>>
>>
>>
>> Bingo! We now know that the handicap for the runner
>> is 6.8 at n = 0 by information reaped in a later moment
>> in time when n = 2... Three frames later.
>>
>>
>> Is this Kosher?!?!
>>
>>
>>
>> :^o
>>
>> ____________________________
>
> If you add zero to .999 repeating you still get .999 repeating.
> Add the infinitely small and you get 1 instead.

torsdag 26 maj 2022 kl. 22:47:38 UTC+2 skrev mitchr...@gmail.com:
> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> > On 5/26/2022 1:25 PM, Michael Moroney wrote:
> > > On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> > >> On 5/25/2022 7:21 PM, Dan joyce wrote:
> > >>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> > >>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> > >>>>> Just because in theory an infinite number of steps is required doesn't
> > >>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> > >>>>> Achilles
> > >>>>> races a tortoise with a head start. Each time Achilles reaches a point
> > >>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> > >>>>> reaches that point, the tortoise advances more. And so on for an
> > >>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> > >>>>> passes it and wins the race, despite taking an infinite number of
> > >>>>> steps
> > >>>>> to catch up to the tortoise.
> > >>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> > >>>> basis:
> > >
> > > What are "the tortoise's rules"? The only rules are the tortoise gets a
> > > head start and both it and Achilles run as fast as they can to the
> > > finish line, and whoever does so first, wins.
> > >>>>
> > >>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> > >>>> tortoise
> > >>>> is ahead because of the head start.
> > >>>>
> > >>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> > >>>> tortoise
> > >>>> is still ahead because of the head start.
> > >>>>
> > >>>> on and on. The turtle will cross the finish line before Achilles.
> > >>>
> > >>> The turtle will never cross the finish line but will always be ahead
> > >>> of Achilles.
> > >>
> > >> Yes. True. It gets infinitely closer and closer to the finish line.
> > >
> > > That is not Zeno's Paradox. The tortoise gets a head start, at point
> > > A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> > > moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> > > the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> > > forth.
> > >
> > > Since Achilles is faster than the tortoise, the distances A[1], A[2],
> > > [A3], ... get smaller and smaller, since the time it takes Achilles to
> > > run from the start to A[1] equals the time it takes the slower tortoise
> > > to run from A[1] to A[2], and so on.
> > >
> > > The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> > > always behind A[n+1] (the tortoise's position), even as n approaches
> > > infinity. So Achilles can never beat the tortoise, right? But, as long
> > > as the head start isn't _too_ large, in real life, Achilles passes the
> > > tortoise and wins, just as you'd expect. So what's wrong with this?
> > >
> > > As I said, just because there's an infinite limit, it doesn't mean the
> > > limit is absolute. In this case, the total time passed also reaches a
> > > limit (at n=infinity) but that time limit isn't infinite, so what
> > > happens after the "limit" on time passes? As always, time marches on...
> > > At that point Achilles passes the tortoise and remains ahead for the
> > > rest of the race, and the infinite series no longer applies.
> > >>
> > >>
> > >>>> Now, if Achilles tells the tortoise to f-off and just starts
> > >>>> running, he
> > >>>> will quickly pass the tortoise...
> > >
> > > In real life, yes, but in Zeno's Paradox, no.
> > >>>>
> > >>>> ;^)
> > >>
> > >
> > > Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> > > (S1>S2), and the head start distance A1, how long does it take for
> > > Achilles to pass the tortoise? :-)
> > >
> > I did some equations on this a while back:
> >
> > https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> >
> > Here are my comments:
> >
> > Iirc, scale was speed:
> > ____________________________
> > [...]
> > Ahhhh, now this is a direct formula:
> >
> > n = iteration count
> > d = distance
> > s = scale
> >
> > r_[n] = (d / s^n) * (s^n - (s-1)^n)
> >
> >
> > just might work for finding the total distance
> > traveled at a given iteration count of the following
> > iterated equation:
> >
> > r_[n+1] = r_[n] + (d - r_[n]) / s
> >
> >
> >
> > Here is the sequence for d = 10 and s = 4 using the
> > iterative formula:
> > __________________________________
> > r_[0] = 0
> > r_[1] = 0 + (10 - 0) / 4 = 2.5
> > r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> > r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> > r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> > __________________________________
> >
> >
> > And here is the sequence for d = 10 and s = 4 using
> > the direct formula:
> > __________________________________
> > r_[0] = 10 / 1 * 0 = 0
> > r_[1] = 10 / 4 * 1 = 2.5
> > r_[2] = 10 / 16 * 7 = 4.375
> > r_[3] = 10 / 64 * 37 = 5.78125
> > r_[4] = 10 / 256 * 175 = 6.8359375
> > __________________________________
> >
> >
> > As you can see, they are identical!
> >
> > Humm...
> > ____________________________
> >
> >
> > Here is another post:
> >
> > https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> > ____________________________
> > I think I found a way to find the handicap of a
> > runner in an infinite race on a finite track...
> >
> > How about something like:
> >
> >
> > Let:
> >
> > d = total distance in track
> > s = scale, which relates to speed
> > n = integer iteration count, which relates to time
> > r_h = a runners starting handicap
> >
> >
> >
> > Here is the iterative equation for finding the
> > distance a runner is down the track that I posted
> > up thread:
> >
> > r_[n + 1] = r_[n] + (d - r_[n]) / s
> >
> >
> > The handicap of the runner is equal to r_[0]
> > because n = 0 is the starting position of every
> > runner.
> >
> > The goal is to find the handicap of a runner with
> > a given distance, iteration count, total distance
> > of the track, and a scale or speed. AFAICT, the
> > following formula solves for the handicap of a
> > runner using that information:
> >
> >
> > r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> >
> >
> >
> > Here is output of a racer using the iterative equation
> > with the following attributes:
> >
> > d = 10
> > s = 4
> > r_h = 6.8
> > _______________________________________
> > r_[0] = 6.8
> > r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> > r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> > r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> > r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> > _______________________________________
> >
> >
> >
> > As we can see this runner has a head start of 6.8 out
> > of 10. Also, in the third frame, the runner r_[2] has
> > traveled 8.2 out of a possible 10.0.
> >
> > Given that information alone, we can plug it all into
> > the formula for finding the handicap, and get:
> >
> >
> > r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> >
> >
> >
> > Bingo! We now know that the handicap for the runner
> > is 6.8 at n = 0 by information reaped in a later moment
> > in time when n = 2... Three frames later.
> >
> >
> > Is this Kosher?!?!
> >
> >
> >
> > :^o
> >
> > ____________________________
> If you add zero to .999 repeating you still get .999 repeating.
> Add the infinitely small and you get 1 instead.

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.

On Thursday, May 26, 2022 at 1:25:20 PM UTC-7, Michael Moroney wrote:
> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> > On 5/25/2022 7:21 PM, Dan joyce wrote:
> >> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> >>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> >>>> Just because in theory an infinite number of steps is required doesn't
> >>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> >>>> Achilles
> >>>> races a tortoise with a head start. Each time Achilles reaches a point
> >>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> >>>> reaches that point, the tortoise advances more. And so on for an
> >>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> >>>> passes it and wins the race, despite taking an infinite number of steps
> >>>> to catch up to the tortoise.
> >>> If Achilles strictly plays by the tortoises rules on a step-by-step
> >>> basis:
> What are "the tortoise's rules"? The only rules are the tortoise gets a
> head start and both it and Achilles run as fast as they can to the
> finish line, and whoever does so first, wins.
> >>>
> >>> step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
> >>> is ahead because of the head start.
> >>>
> >>> step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
> >>> is still ahead because of the head start.
> >>>
> >>> on and on. The turtle will cross the finish line before Achilles.
> >>
> >> The turtle will never cross the finish line but will always be ahead
> >> of Achilles.
> >
> > Yes. True. It gets infinitely closer and closer to the finish line.
> That is not Zeno's Paradox. The tortoise gets a head start, at point
> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so forth.
>
> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> [A3], ... get smaller and smaller, since the time it takes Achilles to
> run from the start to A[1] equals the time it takes the slower tortoise
> to run from A[1] to A[2], and so on.
>
> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> always behind A[n+1] (the tortoise's position), even as n approaches
> infinity. So Achilles can never beat the tortoise, right? But, as long
> as the head start isn't _too_ large, in real life, Achilles passes the
> tortoise and wins, just as you'd expect. So what's wrong with this?
>
> As I said, just because there's an infinite limit, it doesn't mean the
> limit is absolute. In this case, the total time passed also reaches a
> limit (at n=infinity) but that time limit isn't infinite, so what
> happens after the "limit" on time passes? As always, time marches on...
> At that point Achilles passes the tortoise and remains ahead for the
> rest of the race, and the infinite series no longer applies.
> >
> >
> >>> Now, if Achilles tells the tortoise to f-off and just starts running, he
> >>> will quickly pass the tortoise...
> In real life, yes, but in Zeno's Paradox, no.
> >>>
> >>> ;^)
> >
>
> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> (S1>S2), and the head start distance A1, how long does it take for
> Achilles to pass the tortoise? :-)

It's like, what is the turtle's track, on the inside, and Achilles, on
the outside, that they run the same track, it's the perimeter.

For whatever ratio in velocity Achilles to turtle, the radius difference
is half for "pi" times diameter the perimeter, while, the area covered,
is a function of time. Length and area are both parameterized in t.
Then, by what "area" Achilles covers in time, when he arrives at the
finish, the turtle will have only covered so much area.

So, by measuring them in time first the ratio to determin the
ratio of Achilles to tortoise their velocity, which is greater-than-1,
it draws that Achilles finishes the race on the straight track,
so ahead of the turtle.

Then the case where they finish at the same time is where
Achilles passes the tortoise.

On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
> On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> > On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> >> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> >>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> >>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
> >>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> >>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> >>>>>>> Just because in theory an infinite number of steps is required doesn't
> >>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> >>>>>>> Achilles
> >>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
> >>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> >>>>>>> reaches that point, the tortoise advances more. And so on for an
> >>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> >>>>>>> passes it and wins the race, despite taking an infinite number of
> >>>>>>> steps
> >>>>>>> to catch up to the tortoise.
> >>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> >>>>>> basis:
> >>>
> >>> What are "the tortoise's rules"? The only rules are the tortoise gets a
> >>> head start and both it and Achilles run as fast as they can to the
> >>> finish line, and whoever does so first, wins.
> >>>>>>
> >>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> >>>>>> tortoise
> >>>>>> is ahead because of the head start.
> >>>>>>
> >>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> >>>>>> tortoise
> >>>>>> is still ahead because of the head start.
> >>>>>>
> >>>>>> on and on. The turtle will cross the finish line before Achilles.
> >>>>>
> >>>>> The turtle will never cross the finish line but will always be ahead
> >>>>> of Achilles.
> >>>>
> >>>> Yes. True. It gets infinitely closer and closer to the finish line.
> >>>
> >>> That is not Zeno's Paradox. The tortoise gets a head start, at point
> >>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> >>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> >>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> >>> forth.
> >>>
> >>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> >>> [A3], ... get smaller and smaller, since the time it takes Achilles to
> >>> run from the start to A[1] equals the time it takes the slower tortoise
> >>> to run from A[1] to A[2], and so on.
> >>>
> >>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> >>> always behind A[n+1] (the tortoise's position), even as n approaches
> >>> infinity. So Achilles can never beat the tortoise, right? But, as long
> >>> as the head start isn't _too_ large, in real life, Achilles passes the
> >>> tortoise and wins, just as you'd expect. So what's wrong with this?
> >>>
> >>> As I said, just because there's an infinite limit, it doesn't mean the
> >>> limit is absolute. In this case, the total time passed also reaches a
> >>> limit (at n=infinity) but that time limit isn't infinite, so what
> >>> happens after the "limit" on time passes? As always, time marches on...
> >>> At that point Achilles passes the tortoise and remains ahead for the
> >>> rest of the race, and the infinite series no longer applies.
> >>>>
> >>>>
> >>>>>> Now, if Achilles tells the tortoise to f-off and just starts
> >>>>>> running, he
> >>>>>> will quickly pass the tortoise...
> >>>
> >>> In real life, yes, but in Zeno's Paradox, no.
> >>>>>>
> >>>>>> ;^)
> >>>>
> >>>
> >>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> >>> (S1>S2), and the head start distance A1, how long does it take for
> >>> Achilles to pass the tortoise? :-)
> >>>
> >> I did some equations on this a while back:
> >>
> >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> >>
> >> Here are my comments:
> >>
> >> Iirc, scale was speed:
> >> ____________________________
> >> [...]
> >> Ahhhh, now this is a direct formula:
> >>
> >> n = iteration count
> >> d = distance
> >> s = scale
> >>
> >> r_[n] = (d / s^n) * (s^n - (s-1)^n)
> >>
> >>
> >> just might work for finding the total distance
> >> traveled at a given iteration count of the following
> >> iterated equation:
> >>
> >> r_[n+1] = r_[n] + (d - r_[n]) / s
> >>
> >>
> >>
> >> Here is the sequence for d = 10 and s = 4 using the
> >> iterative formula:
> >> __________________________________
> >> r_[0] = 0
> >> r_[1] = 0 + (10 - 0) / 4 = 2.5
> >> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> >> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> >> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> >> __________________________________
> >>
> >>
> >> And here is the sequence for d = 10 and s = 4 using
> >> the direct formula:
> >> __________________________________
> >> r_[0] = 10 / 1 * 0 = 0
> >> r_[1] = 10 / 4 * 1 = 2.5
> >> r_[2] = 10 / 16 * 7 = 4.375
> >> r_[3] = 10 / 64 * 37 = 5.78125
> >> r_[4] = 10 / 256 * 175 = 6.8359375
> >> __________________________________
> >>
> >>
> >> As you can see, they are identical!
> >>
> >> Humm...
> >> ____________________________
> >>
> >>
> >> Here is another post:
> >>
> >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> >> ____________________________
> >> I think I found a way to find the handicap of a
> >> runner in an infinite race on a finite track...
> >>
> >> How about something like:
> >>
> >>
> >> Let:
> >>
> >> d = total distance in track
> >> s = scale, which relates to speed
> >> n = integer iteration count, which relates to time
> >> r_h = a runners starting handicap
> >>
> >>
> >>
> >> Here is the iterative equation for finding the
> >> distance a runner is down the track that I posted
> >> up thread:
> >>
> >> r_[n + 1] = r_[n] + (d - r_[n]) / s
> >>
> >>
> >> The handicap of the runner is equal to r_[0]
> >> because n = 0 is the starting position of every
> >> runner.
> >>
> >> The goal is to find the handicap of a runner with
> >> a given distance, iteration count, total distance
> >> of the track, and a scale or speed. AFAICT, the
> >> following formula solves for the handicap of a
> >> runner using that information:
> >>
> >>
> >> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> >>
> >>
> >>
> >> Here is output of a racer using the iterative equation
> >> with the following attributes:
> >>
> >> d = 10
> >> s = 4
> >> r_h = 6.8
> >> _______________________________________
> >> r_[0] = 6.8
> >> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> >> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> >> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> >> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> >> _______________________________________
> >>
> >>
> >>
> >> As we can see this runner has a head start of 6.8 out
> >> of 10. Also, in the third frame, the runner r_[2] has
> >> traveled 8.2 out of a possible 10.0.
> >>
> >> Given that information alone, we can plug it all into
> >> the formula for finding the handicap, and get:
> >>
> >>
> >> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> >>
> >>
> >>
> >> Bingo! We now know that the handicap for the runner
> >> is 6.8 at n = 0 by information reaped in a later moment
> >> in time when n = 2... Three frames later.
> >>
> >>
> >> Is this Kosher?!?!
> >>
> >>
> >>
> >> :^o
> >>
> >> ____________________________
> >
> > If you add zero to .999 repeating you still get .999 repeating.
> > Add the infinitely small and you get 1 instead.
> .999 repeating = 1.000 repeating anyway

On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
> On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
> > On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> > > On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> > >> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> > >>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> > >>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
> > >>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> > >>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> > >>>>>>> Just because in theory an infinite number of steps is required doesn't
> > >>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> > >>>>>>> Achilles
> > >>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
> > >>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> > >>>>>>> reaches that point, the tortoise advances more. And so on for an
> > >>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> > >>>>>>> passes it and wins the race, despite taking an infinite number of
> > >>>>>>> steps
> > >>>>>>> to catch up to the tortoise.
> > >>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> > >>>>>> basis:
> > >>>
> > >>> What are "the tortoise's rules"? The only rules are the tortoise gets a
> > >>> head start and both it and Achilles run as fast as they can to the
> > >>> finish line, and whoever does so first, wins.
> > >>>>>>
> > >>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> > >>>>>> tortoise
> > >>>>>> is ahead because of the head start.
> > >>>>>>
> > >>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> > >>>>>> tortoise
> > >>>>>> is still ahead because of the head start.
> > >>>>>>
> > >>>>>> on and on. The turtle will cross the finish line before Achilles.
> > >>>>>
> > >>>>> The turtle will never cross the finish line but will always be ahead
> > >>>>> of Achilles.
> > >>>>
> > >>>> Yes. True. It gets infinitely closer and closer to the finish line.
> > >>>
> > >>> That is not Zeno's Paradox. The tortoise gets a head start, at point
> > >>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> > >>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> > >>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> > >>> forth.
> > >>>
> > >>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> > >>> [A3], ... get smaller and smaller, since the time it takes Achilles to
> > >>> run from the start to A[1] equals the time it takes the slower tortoise
> > >>> to run from A[1] to A[2], and so on.
> > >>>
> > >>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> > >>> always behind A[n+1] (the tortoise's position), even as n approaches
> > >>> infinity. So Achilles can never beat the tortoise, right? But, as long
> > >>> as the head start isn't _too_ large, in real life, Achilles passes the
> > >>> tortoise and wins, just as you'd expect. So what's wrong with this?
> > >>>
> > >>> As I said, just because there's an infinite limit, it doesn't mean the
> > >>> limit is absolute. In this case, the total time passed also reaches a
> > >>> limit (at n=infinity) but that time limit isn't infinite, so what
> > >>> happens after the "limit" on time passes? As always, time marches on...
> > >>> At that point Achilles passes the tortoise and remains ahead for the
> > >>> rest of the race, and the infinite series no longer applies.
> > >>>>
> > >>>>
> > >>>>>> Now, if Achilles tells the tortoise to f-off and just starts
> > >>>>>> running, he
> > >>>>>> will quickly pass the tortoise...
> > >>>
> > >>> In real life, yes, but in Zeno's Paradox, no.
> > >>>>>>
> > >>>>>> ;^)
> > >>>>
> > >>>
> > >>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> > >>> (S1>S2), and the head start distance A1, how long does it take for
> > >>> Achilles to pass the tortoise? :-)
> > >>>
> > >> I did some equations on this a while back:
> > >>
> > >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> > >>
> > >> Here are my comments:
> > >>
> > >> Iirc, scale was speed:
> > >> ____________________________
> > >> [...]
> > >> Ahhhh, now this is a direct formula:
> > >>
> > >> n = iteration count
> > >> d = distance
> > >> s = scale
> > >>
> > >> r_[n] = (d / s^n) * (s^n - (s-1)^n)
> > >>
> > >>
> > >> just might work for finding the total distance
> > >> traveled at a given iteration count of the following
> > >> iterated equation:
> > >>
> > >> r_[n+1] = r_[n] + (d - r_[n]) / s
> > >>
> > >>
> > >>
> > >> Here is the sequence for d = 10 and s = 4 using the
> > >> iterative formula:
> > >> __________________________________
> > >> r_[0] = 0
> > >> r_[1] = 0 + (10 - 0) / 4 = 2.5
> > >> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> > >> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> > >> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> > >> __________________________________
> > >>
> > >>
> > >> And here is the sequence for d = 10 and s = 4 using
> > >> the direct formula:
> > >> __________________________________
> > >> r_[0] = 10 / 1 * 0 = 0
> > >> r_[1] = 10 / 4 * 1 = 2.5
> > >> r_[2] = 10 / 16 * 7 = 4.375
> > >> r_[3] = 10 / 64 * 37 = 5.78125
> > >> r_[4] = 10 / 256 * 175 = 6.8359375
> > >> __________________________________
> > >>
> > >>
> > >> As you can see, they are identical!
> > >>
> > >> Humm...
> > >> ____________________________
> > >>
> > >>
> > >> Here is another post:
> > >>
> > >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> > >> ____________________________
> > >> I think I found a way to find the handicap of a
> > >> runner in an infinite race on a finite track...
> > >>
> > >> How about something like:
> > >>
> > >>
> > >> Let:
> > >>
> > >> d = total distance in track
> > >> s = scale, which relates to speed
> > >> n = integer iteration count, which relates to time
> > >> r_h = a runners starting handicap
> > >>
> > >>
> > >>
> > >> Here is the iterative equation for finding the
> > >> distance a runner is down the track that I posted
> > >> up thread:
> > >>
> > >> r_[n + 1] = r_[n] + (d - r_[n]) / s
> > >>
> > >>
> > >> The handicap of the runner is equal to r_[0]
> > >> because n = 0 is the starting position of every
> > >> runner.
> > >>
> > >> The goal is to find the handicap of a runner with
> > >> a given distance, iteration count, total distance
> > >> of the track, and a scale or speed. AFAICT, the
> > >> following formula solves for the handicap of a
> > >> runner using that information:
> > >>
> > >>
> > >> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> > >>
> > >>
> > >>
> > >> Here is output of a racer using the iterative equation
> > >> with the following attributes:
> > >>
> > >> d = 10
> > >> s = 4
> > >> r_h = 6.8
> > >> _______________________________________
> > >> r_[0] = 6.8
> > >> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> > >> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> > >> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> > >> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> > >> _______________________________________
> > >>
> > >>
> > >>
> > >> As we can see this runner has a head start of 6.8 out
> > >> of 10. Also, in the third frame, the runner r_[2] has
> > >> traveled 8.2 out of a possible 10.0.
> > >>
> > >> Given that information alone, we can plug it all into
> > >> the formula for finding the handicap, and get:
> > >>
> > >>
> > >> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> > >>
> > >>
> > >>
> > >> Bingo! We now know that the handicap for the runner
> > >> is 6.8 at n = 0 by information reaped in a later moment
> > >> in time when n = 2... Three frames later.
> > >>
> > >>
> > >> Is this Kosher?!?!
> > >>
> > >>
> > >>
> > >> :^o
> > >>
> > >> ____________________________
> > >
> > > If you add zero to .999 repeating you still get .999 repeating.
> > > Add the infinitely small and you get 1 instead.
> > .999 repeating = 1.000 repeating anyway
> Mitch, for that ".999... is add infinitesimal", just first
> have it that "1 minus infinitesimal, is, .999..., lesser".

On 5/27/2022 12:38 PM, mitchr...@gmail.com wrote:
> On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
>> On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
>>> On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
>>>> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
>>>>> On 5/26/2022 1:25 PM, Michael Moroney wrote:
>>>>>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
>>>>>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
>>>>>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>>>>>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>>>>>>>>> Just because in theory an infinite number of steps is required doesn't
>>>>>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
>>>>>>>>>> Achilles
>>>>>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
>>>>>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>>>>>>>>> reaches that point, the tortoise advances more. And so on for an
>>>>>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
>>>>>>>>>> passes it and wins the race, despite taking an infinite number of
>>>>>>>>>> steps
>>>>>>>>>> to catch up to the tortoise.
>>>>>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
>>>>>>>>> basis:
>>>>>>
>>>>>> What are "the tortoise's rules"? The only rules are the tortoise gets a
>>>>>> head start and both it and Achilles run as fast as they can to the
>>>>>> finish line, and whoever does so first, wins.
>>>>>>>>>
>>>>>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
>>>>>>>>> tortoise
>>>>>>>>> is ahead because of the head start.
>>>>>>>>>
>>>>>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
>>>>>>>>> tortoise
>>>>>>>>> is still ahead because of the head start.
>>>>>>>>>
>>>>>>>>> on and on. The turtle will cross the finish line before Achilles.
>>>>>>>>
>>>>>>>> The turtle will never cross the finish line but will always be ahead
>>>>>>>> of Achilles.
>>>>>>>
>>>>>>> Yes. True. It gets infinitely closer and closer to the finish line.
>>>>>>
>>>>>> That is not Zeno's Paradox. The tortoise gets a head start, at point
>>>>>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
>>>>>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
>>>>>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
>>>>>> forth.
>>>>>>
>>>>>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
>>>>>> [A3], ... get smaller and smaller, since the time it takes Achilles to
>>>>>> run from the start to A[1] equals the time it takes the slower tortoise
>>>>>> to run from A[1] to A[2], and so on.
>>>>>>
>>>>>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
>>>>>> always behind A[n+1] (the tortoise's position), even as n approaches
>>>>>> infinity. So Achilles can never beat the tortoise, right? But, as long
>>>>>> as the head start isn't _too_ large, in real life, Achilles passes the
>>>>>> tortoise and wins, just as you'd expect. So what's wrong with this?
>>>>>>
>>>>>> As I said, just because there's an infinite limit, it doesn't mean the
>>>>>> limit is absolute. In this case, the total time passed also reaches a
>>>>>> limit (at n=infinity) but that time limit isn't infinite, so what
>>>>>> happens after the "limit" on time passes? As always, time marches on...
>>>>>> At that point Achilles passes the tortoise and remains ahead for the
>>>>>> rest of the race, and the infinite series no longer applies.
>>>>>>>
>>>>>>>
>>>>>>>>> Now, if Achilles tells the tortoise to f-off and just starts
>>>>>>>>> running, he
>>>>>>>>> will quickly pass the tortoise...
>>>>>>
>>>>>> In real life, yes, but in Zeno's Paradox, no.
>>>>>>>>>
>>>>>>>>> ;^)
>>>>>>>
>>>>>>
>>>>>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
>>>>>> (S1>S2), and the head start distance A1, how long does it take for
>>>>>> Achilles to pass the tortoise? :-)
>>>>>>
>>>>> I did some equations on this a while back:
>>>>>
>>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
>>>>>
>>>>> Here are my comments:
>>>>>
>>>>> Iirc, scale was speed:
>>>>> ____________________________
>>>>> [...]
>>>>> Ahhhh, now this is a direct formula:
>>>>>
>>>>> n = iteration count
>>>>> d = distance
>>>>> s = scale
>>>>>
>>>>> r_[n] = (d / s^n) * (s^n - (s-1)^n)
>>>>>
>>>>>
>>>>> just might work for finding the total distance
>>>>> traveled at a given iteration count of the following
>>>>> iterated equation:
>>>>>
>>>>> r_[n+1] = r_[n] + (d - r_[n]) / s
>>>>>
>>>>>
>>>>>
>>>>> Here is the sequence for d = 10 and s = 4 using the
>>>>> iterative formula:
>>>>> __________________________________
>>>>> r_[0] = 0
>>>>> r_[1] = 0 + (10 - 0) / 4 = 2.5
>>>>> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
>>>>> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
>>>>> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
>>>>> __________________________________
>>>>>
>>>>>
>>>>> And here is the sequence for d = 10 and s = 4 using
>>>>> the direct formula:
>>>>> __________________________________
>>>>> r_[0] = 10 / 1 * 0 = 0
>>>>> r_[1] = 10 / 4 * 1 = 2.5
>>>>> r_[2] = 10 / 16 * 7 = 4.375
>>>>> r_[3] = 10 / 64 * 37 = 5.78125
>>>>> r_[4] = 10 / 256 * 175 = 6.8359375
>>>>> __________________________________
>>>>>
>>>>>
>>>>> As you can see, they are identical!
>>>>>
>>>>> Humm...
>>>>> ____________________________
>>>>>
>>>>>
>>>>> Here is another post:
>>>>>
>>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
>>>>> ____________________________
>>>>> I think I found a way to find the handicap of a
>>>>> runner in an infinite race on a finite track...
>>>>>
>>>>> How about something like:
>>>>>
>>>>>
>>>>> Let:
>>>>>
>>>>> d = total distance in track
>>>>> s = scale, which relates to speed
>>>>> n = integer iteration count, which relates to time
>>>>> r_h = a runners starting handicap
>>>>>
>>>>>
>>>>>
>>>>> Here is the iterative equation for finding the
>>>>> distance a runner is down the track that I posted
>>>>> up thread:
>>>>>
>>>>> r_[n + 1] = r_[n] + (d - r_[n]) / s
>>>>>
>>>>>
>>>>> The handicap of the runner is equal to r_[0]
>>>>> because n = 0 is the starting position of every
>>>>> runner.
>>>>>
>>>>> The goal is to find the handicap of a runner with
>>>>> a given distance, iteration count, total distance
>>>>> of the track, and a scale or speed. AFAICT, the
>>>>> following formula solves for the handicap of a
>>>>> runner using that information:
>>>>>
>>>>>
>>>>> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
>>>>>
>>>>>
>>>>>
>>>>> Here is output of a racer using the iterative equation
>>>>> with the following attributes:
>>>>>
>>>>> d = 10
>>>>> s = 4
>>>>> r_h = 6.8
>>>>> _______________________________________
>>>>> r_[0] = 6.8
>>>>> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
>>>>> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
>>>>> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
>>>>> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
>>>>> _______________________________________
>>>>>
>>>>>
>>>>>
>>>>> As we can see this runner has a head start of 6.8 out
>>>>> of 10. Also, in the third frame, the runner r_[2] has
>>>>> traveled 8.2 out of a possible 10.0.
>>>>>
>>>>> Given that information alone, we can plug it all into
>>>>> the formula for finding the handicap, and get:
>>>>>
>>>>>
>>>>> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
>>>>>
>>>>>
>>>>>
>>>>> Bingo! We now know that the handicap for the runner
>>>>> is 6.8 at n = 0 by information reaped in a later moment
>>>>> in time when n = 2... Three frames later.
>>>>>
>>>>>
>>>>> Is this Kosher?!?!
>>>>>
>>>>>
>>>>>
>>>>> :^o
>>>>>
>>>>> ____________________________
>>>>
>>>> If you add zero to .999 repeating you still get .999 repeating.
>>>> Add the infinitely small and you get 1 instead.
>>> .999 repeating = 1.000 repeating anyway
>> Mitch, for that ".999... is add infinitesimal", just first
>> have it that "1 minus infinitesimal, is, .999..., lesser".
>
> .999 is lesser than one by the infinitely small not zero.
>
> Mitchell Raemsch

Michael Moroney wrote:

> Just because in theory an infinite number of steps is required doesn't
> mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
> races a tortoise with a head start. Each time Achilles reaches a point
> where the tortoise was, the tortoise advances somewhat. When Achilles
> reaches that point, the tortoise advances more. And so on for an
> infinite number of steps. Yet Achilles catches up to the tortoise and
> passes it and wins the race, despite taking an infinite number of steps
> to catch up to the tortoise.

"Ukrainian President Volodymyr Zelensky, foreground, and Polish President
Andrzej Duda pictured in Kiev, Ukraine on May 22, 2022. © AP / Efrem
Lukatsky"
https://www.rt.com/russia/556220-ukraine-poland-merger-yanukovich/

"Duda", what on earth *nazi_name* is that, Duda??

On Friday, May 27, 2022 at 11:21:08 AM UTC-7, sergi o wrote:
> On 5/27/2022 12:38 PM, mitchr...@gmail.com wrote:
> > On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
> >> On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
> >>> On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> >>>> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> >>>>> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> >>>>>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> >>>>>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
> >>>>>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> >>>>>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> >>>>>>>>>> Just because in theory an infinite number of steps is required doesn't
> >>>>>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> >>>>>>>>>> Achilles
> >>>>>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
> >>>>>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> >>>>>>>>>> reaches that point, the tortoise advances more. And so on for an
> >>>>>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> >>>>>>>>>> passes it and wins the race, despite taking an infinite number of
> >>>>>>>>>> steps
> >>>>>>>>>> to catch up to the tortoise.
> >>>>>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> >>>>>>>>> basis:
> >>>>>>
> >>>>>> What are "the tortoise's rules"? The only rules are the tortoise gets a
> >>>>>> head start and both it and Achilles run as fast as they can to the
> >>>>>> finish line, and whoever does so first, wins.
> >>>>>>>>>
> >>>>>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> >>>>>>>>> tortoise
> >>>>>>>>> is ahead because of the head start.
> >>>>>>>>>
> >>>>>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> >>>>>>>>> tortoise
> >>>>>>>>> is still ahead because of the head start.
> >>>>>>>>>
> >>>>>>>>> on and on. The turtle will cross the finish line before Achilles.
> >>>>>>>>
> >>>>>>>> The turtle will never cross the finish line but will always be ahead
> >>>>>>>> of Achilles.
> >>>>>>>
> >>>>>>> Yes. True. It gets infinitely closer and closer to the finish line.
> >>>>>>
> >>>>>> That is not Zeno's Paradox. The tortoise gets a head start, at point
> >>>>>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> >>>>>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> >>>>>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> >>>>>> forth.
> >>>>>>
> >>>>>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> >>>>>> [A3], ... get smaller and smaller, since the time it takes Achilles to
> >>>>>> run from the start to A[1] equals the time it takes the slower tortoise
> >>>>>> to run from A[1] to A[2], and so on.
> >>>>>>
> >>>>>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> >>>>>> always behind A[n+1] (the tortoise's position), even as n approaches
> >>>>>> infinity. So Achilles can never beat the tortoise, right? But, as long
> >>>>>> as the head start isn't _too_ large, in real life, Achilles passes the
> >>>>>> tortoise and wins, just as you'd expect. So what's wrong with this?
> >>>>>>
> >>>>>> As I said, just because there's an infinite limit, it doesn't mean the
> >>>>>> limit is absolute. In this case, the total time passed also reaches a
> >>>>>> limit (at n=infinity) but that time limit isn't infinite, so what
> >>>>>> happens after the "limit" on time passes? As always, time marches on...
> >>>>>> At that point Achilles passes the tortoise and remains ahead for the
> >>>>>> rest of the race, and the infinite series no longer applies.
> >>>>>>>
> >>>>>>>
> >>>>>>>>> Now, if Achilles tells the tortoise to f-off and just starts
> >>>>>>>>> running, he
> >>>>>>>>> will quickly pass the tortoise...
> >>>>>>
> >>>>>> In real life, yes, but in Zeno's Paradox, no.
> >>>>>>>>>
> >>>>>>>>> ;^)
> >>>>>>>
> >>>>>>
> >>>>>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> >>>>>> (S1>S2), and the head start distance A1, how long does it take for
> >>>>>> Achilles to pass the tortoise? :-)
> >>>>>>
> >>>>> I did some equations on this a while back:
> >>>>>
> >>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> >>>>>
> >>>>> Here are my comments:
> >>>>>
> >>>>> Iirc, scale was speed:
> >>>>> ____________________________
> >>>>> [...]
> >>>>> Ahhhh, now this is a direct formula:
> >>>>>
> >>>>> n = iteration count
> >>>>> d = distance
> >>>>> s = scale
> >>>>>
> >>>>> r_[n] = (d / s^n) * (s^n - (s-1)^n)
> >>>>>
> >>>>>
> >>>>> just might work for finding the total distance
> >>>>> traveled at a given iteration count of the following
> >>>>> iterated equation:
> >>>>>
> >>>>> r_[n+1] = r_[n] + (d - r_[n]) / s
> >>>>>
> >>>>>
> >>>>>
> >>>>> Here is the sequence for d = 10 and s = 4 using the
> >>>>> iterative formula:
> >>>>> __________________________________
> >>>>> r_[0] = 0
> >>>>> r_[1] = 0 + (10 - 0) / 4 = 2.5
> >>>>> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> >>>>> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> >>>>> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> >>>>> __________________________________
> >>>>>
> >>>>>
> >>>>> And here is the sequence for d = 10 and s = 4 using
> >>>>> the direct formula:
> >>>>> __________________________________
> >>>>> r_[0] = 10 / 1 * 0 = 0
> >>>>> r_[1] = 10 / 4 * 1 = 2.5
> >>>>> r_[2] = 10 / 16 * 7 = 4.375
> >>>>> r_[3] = 10 / 64 * 37 = 5.78125
> >>>>> r_[4] = 10 / 256 * 175 = 6.8359375
> >>>>> __________________________________
> >>>>>
> >>>>>
> >>>>> As you can see, they are identical!
> >>>>>
> >>>>> Humm...
> >>>>> ____________________________
> >>>>>
> >>>>>
> >>>>> Here is another post:
> >>>>>
> >>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> >>>>> ____________________________
> >>>>> I think I found a way to find the handicap of a
> >>>>> runner in an infinite race on a finite track...
> >>>>>
> >>>>> How about something like:
> >>>>>
> >>>>>
> >>>>> Let:
> >>>>>
> >>>>> d = total distance in track
> >>>>> s = scale, which relates to speed
> >>>>> n = integer iteration count, which relates to time
> >>>>> r_h = a runners starting handicap
> >>>>>
> >>>>>
> >>>>>
> >>>>> Here is the iterative equation for finding the
> >>>>> distance a runner is down the track that I posted
> >>>>> up thread:
> >>>>>
> >>>>> r_[n + 1] = r_[n] + (d - r_[n]) / s
> >>>>>
> >>>>>
> >>>>> The handicap of the runner is equal to r_[0]
> >>>>> because n = 0 is the starting position of every
> >>>>> runner.
> >>>>>
> >>>>> The goal is to find the handicap of a runner with
> >>>>> a given distance, iteration count, total distance
> >>>>> of the track, and a scale or speed. AFAICT, the
> >>>>> following formula solves for the handicap of a
> >>>>> runner using that information:
> >>>>>
> >>>>>
> >>>>> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> >>>>>
> >>>>>
> >>>>>
> >>>>> Here is output of a racer using the iterative equation
> >>>>> with the following attributes:
> >>>>>
> >>>>> d = 10
> >>>>> s = 4
> >>>>> r_h = 6.8
> >>>>> _______________________________________
> >>>>> r_[0] = 6.8
> >>>>> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> >>>>> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> >>>>> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> >>>>> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> >>>>> _______________________________________
> >>>>>
> >>>>>
> >>>>>
> >>>>> As we can see this runner has a head start of 6.8 out
> >>>>> of 10. Also, in the third frame, the runner r_[2] has
> >>>>> traveled 8.2 out of a possible 10.0.
> >>>>>
> >>>>> Given that information alone, we can plug it all into
> >>>>> the formula for finding the handicap, and get:
> >>>>>
> >>>>>
> >>>>> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> >>>>>
> >>>>>
> >>>>>
> >>>>> Bingo! We now know that the handicap for the runner
> >>>>> is 6.8 at n = 0 by information reaped in a later moment
> >>>>> in time when n = 2... Three frames later.
> >>>>>
> >>>>>
> >>>>> Is this Kosher?!?!
> >>>>>
> >>>>>
> >>>>>
> >>>>> :^o
> >>>>>
> >>>>> ____________________________
> >>>>
> >>>> If you add zero to .999 repeating you still get .999 repeating.
> >>>> Add the infinitely small and you get 1 instead.
> >>> .999 repeating = 1.000 repeating anyway
> >> Mitch, for that ".999... is add infinitesimal", just first
> >> have it that "1 minus infinitesimal, is, .999..., lesser".
> >
> > .999 is lesser than one by the infinitely small not zero.
> >
> > Mitchell Raemsch
> if you add an infinitesimal to 1 you get 1.000... repeating

On Friday, May 27, 2022 at 10:38:33 AM UTC-7, mitchr...@gmail.com wrote:
> On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
> > On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
> > > On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> > > > On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> > > >> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> > > >>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> > > >>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
> > > >>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> > > >>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> > > >>>>>>> Just because in theory an infinite number of steps is required doesn't
> > > >>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> > > >>>>>>> Achilles
> > > >>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
> > > >>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> > > >>>>>>> reaches that point, the tortoise advances more. And so on for an
> > > >>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> > > >>>>>>> passes it and wins the race, despite taking an infinite number of
> > > >>>>>>> steps
> > > >>>>>>> to catch up to the tortoise.
> > > >>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> > > >>>>>> basis:
> > > >>>
> > > >>> What are "the tortoise's rules"? The only rules are the tortoise gets a
> > > >>> head start and both it and Achilles run as fast as they can to the
> > > >>> finish line, and whoever does so first, wins.
> > > >>>>>>
> > > >>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> > > >>>>>> tortoise
> > > >>>>>> is ahead because of the head start.
> > > >>>>>>
> > > >>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> > > >>>>>> tortoise
> > > >>>>>> is still ahead because of the head start.
> > > >>>>>>
> > > >>>>>> on and on. The turtle will cross the finish line before Achilles.
> > > >>>>>
> > > >>>>> The turtle will never cross the finish line but will always be ahead
> > > >>>>> of Achilles.
> > > >>>>
> > > >>>> Yes. True. It gets infinitely closer and closer to the finish line.
> > > >>>
> > > >>> That is not Zeno's Paradox. The tortoise gets a head start, at point
> > > >>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> > > >>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> > > >>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> > > >>> forth.
> > > >>>
> > > >>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> > > >>> [A3], ... get smaller and smaller, since the time it takes Achilles to
> > > >>> run from the start to A[1] equals the time it takes the slower tortoise
> > > >>> to run from A[1] to A[2], and so on.
> > > >>>
> > > >>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> > > >>> always behind A[n+1] (the tortoise's position), even as n approaches
> > > >>> infinity. So Achilles can never beat the tortoise, right? But, as long
> > > >>> as the head start isn't _too_ large, in real life, Achilles passes the
> > > >>> tortoise and wins, just as you'd expect. So what's wrong with this?
> > > >>>
> > > >>> As I said, just because there's an infinite limit, it doesn't mean the
> > > >>> limit is absolute. In this case, the total time passed also reaches a
> > > >>> limit (at n=infinity) but that time limit isn't infinite, so what
> > > >>> happens after the "limit" on time passes? As always, time marches on...
> > > >>> At that point Achilles passes the tortoise and remains ahead for the
> > > >>> rest of the race, and the infinite series no longer applies.
> > > >>>>
> > > >>>>
> > > >>>>>> Now, if Achilles tells the tortoise to f-off and just starts
> > > >>>>>> running, he
> > > >>>>>> will quickly pass the tortoise...
> > > >>>
> > > >>> In real life, yes, but in Zeno's Paradox, no.
> > > >>>>>>
> > > >>>>>> ;^)
> > > >>>>
> > > >>>
> > > >>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> > > >>> (S1>S2), and the head start distance A1, how long does it take for
> > > >>> Achilles to pass the tortoise? :-)
> > > >>>
> > > >> I did some equations on this a while back:
> > > >>
> > > >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> > > >>
> > > >> Here are my comments:
> > > >>
> > > >> Iirc, scale was speed:
> > > >> ____________________________
> > > >> [...]
> > > >> Ahhhh, now this is a direct formula:
> > > >>
> > > >> n = iteration count
> > > >> d = distance
> > > >> s = scale
> > > >>
> > > >> r_[n] = (d / s^n) * (s^n - (s-1)^n)
> > > >>
> > > >>
> > > >> just might work for finding the total distance
> > > >> traveled at a given iteration count of the following
> > > >> iterated equation:
> > > >>
> > > >> r_[n+1] = r_[n] + (d - r_[n]) / s
> > > >>
> > > >>
> > > >>
> > > >> Here is the sequence for d = 10 and s = 4 using the
> > > >> iterative formula:
> > > >> __________________________________
> > > >> r_[0] = 0
> > > >> r_[1] = 0 + (10 - 0) / 4 = 2.5
> > > >> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> > > >> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> > > >> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> > > >> __________________________________
> > > >>
> > > >>
> > > >> And here is the sequence for d = 10 and s = 4 using
> > > >> the direct formula:
> > > >> __________________________________
> > > >> r_[0] = 10 / 1 * 0 = 0
> > > >> r_[1] = 10 / 4 * 1 = 2.5
> > > >> r_[2] = 10 / 16 * 7 = 4.375
> > > >> r_[3] = 10 / 64 * 37 = 5.78125
> > > >> r_[4] = 10 / 256 * 175 = 6.8359375
> > > >> __________________________________
> > > >>
> > > >>
> > > >> As you can see, they are identical!
> > > >>
> > > >> Humm...
> > > >> ____________________________
> > > >>
> > > >>
> > > >> Here is another post:
> > > >>
> > > >> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> > > >> ____________________________
> > > >> I think I found a way to find the handicap of a
> > > >> runner in an infinite race on a finite track...
> > > >>
> > > >> How about something like:
> > > >>
> > > >>
> > > >> Let:
> > > >>
> > > >> d = total distance in track
> > > >> s = scale, which relates to speed
> > > >> n = integer iteration count, which relates to time
> > > >> r_h = a runners starting handicap
> > > >>
> > > >>
> > > >>
> > > >> Here is the iterative equation for finding the
> > > >> distance a runner is down the track that I posted
> > > >> up thread:
> > > >>
> > > >> r_[n + 1] = r_[n] + (d - r_[n]) / s
> > > >>
> > > >>
> > > >> The handicap of the runner is equal to r_[0]
> > > >> because n = 0 is the starting position of every
> > > >> runner.
> > > >>
> > > >> The goal is to find the handicap of a runner with
> > > >> a given distance, iteration count, total distance
> > > >> of the track, and a scale or speed. AFAICT, the
> > > >> following formula solves for the handicap of a
> > > >> runner using that information:
> > > >>
> > > >>
> > > >> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> > > >>
> > > >>
> > > >>
> > > >> Here is output of a racer using the iterative equation
> > > >> with the following attributes:
> > > >>
> > > >> d = 10
> > > >> s = 4
> > > >> r_h = 6.8
> > > >> _______________________________________
> > > >> r_[0] = 6.8
> > > >> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> > > >> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> > > >> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> > > >> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> > > >> _______________________________________
> > > >>
> > > >>
> > > >>
> > > >> As we can see this runner has a head start of 6.8 out
> > > >> of 10. Also, in the third frame, the runner r_[2] has
> > > >> traveled 8.2 out of a possible 10.0.
> > > >>
> > > >> Given that information alone, we can plug it all into
> > > >> the formula for finding the handicap, and get:
> > > >>
> > > >>
> > > >> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> > > >>
> > > >>
> > > >>
> > > >> Bingo! We now know that the handicap for the runner
> > > >> is 6.8 at n = 0 by information reaped in a later moment
> > > >> in time when n = 2... Three frames later.
> > > >>
> > > >>
> > > >> Is this Kosher?!?!
> > > >>
> > > >>
> > > >>
> > > >> :^o
> > > >>
> > > >> ____________________________
> > > >
> > > > If you add zero to .999 repeating you still get .999 repeating.
> > > > Add the infinitely small and you get 1 instead.
> > > .999 repeating = 1.000 repeating anyway
> > Mitch, for that ".999... is add infinitesimal", just first
> > have it that "1 minus infinitesimal, is, .999..., lesser".
> .999 is lesser than one by the infinitely small not zero.
>
> Mitchell Raemsch
> >
> > Then though it's always that "the .999..., lesser, is
> > only on its way to zero, least or none", because there
> > are two kinds of relations: related motion and lattice
> > relations, that the field defines lattice relations while
> > the infinitesimals is only part of a "range" or "course".
> >
> > I.e., the infinitesimal changes between 1.0 and 0.0,
> > going through each .aaa... as far as it could be measured,
> > are instead of that "this .333... times 3 = .999... = 1", that
> > this "1 minus .000...1" is writing out a notation, where
> > the ...1's "sum their differences, to zero", while the numbers,
> > "round up".
> >
> > So, when someone writes ".999, ..., repeating", is mostly
> > reflecting the notion that the notation after numbers introducing
> > the "..." or over-bar or the usual way of indicating the
> > repeating part for any rational number, basically works from
> > the field of course that _all_ and _only_ rational numbers,
> > end with a repeating terminus.
> >
> > Then there's only that
> >
> > 000... <- 0
> > 000...
> >
> > 011...
> > 011... <- 1/2
> > 100...
> >
> > 111...
> > 111... <- 1
> >
> > Notice the bounds are only at the ends,
> > and each column is half 1's and half 0's.
> >
> > It's easier to reduce the discussion to [0,1] instead of
> > involving all the real numbers.

On Friday, May 27, 2022 at 4:53:41 PM UTC-7, mitchr...@gmail.com wrote:
> On Friday, May 27, 2022 at 11:21:08 AM UTC-7, sergi o wrote:
> > On 5/27/2022 12:38 PM, mitchr...@gmail.com wrote:
> > > On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
> > >> On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
> > >>> On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
> > >>>> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
> > >>>>> On 5/26/2022 1:25 PM, Michael Moroney wrote:
> > >>>>>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
> > >>>>>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
> > >>>>>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
> > >>>>>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
> > >>>>>>>>>> Just because in theory an infinite number of steps is required doesn't
> > >>>>>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
> > >>>>>>>>>> Achilles
> > >>>>>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
> > >>>>>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
> > >>>>>>>>>> reaches that point, the tortoise advances more. And so on for an
> > >>>>>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
> > >>>>>>>>>> passes it and wins the race, despite taking an infinite number of
> > >>>>>>>>>> steps
> > >>>>>>>>>> to catch up to the tortoise.
> > >>>>>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
> > >>>>>>>>> basis:
> > >>>>>>
> > >>>>>> What are "the tortoise's rules"? The only rules are the tortoise gets a
> > >>>>>> head start and both it and Achilles run as fast as they can to the
> > >>>>>> finish line, and whoever does so first, wins.
> > >>>>>>>>>
> > >>>>>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
> > >>>>>>>>> tortoise
> > >>>>>>>>> is ahead because of the head start.
> > >>>>>>>>>
> > >>>>>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
> > >>>>>>>>> tortoise
> > >>>>>>>>> is still ahead because of the head start.
> > >>>>>>>>>
> > >>>>>>>>> on and on. The turtle will cross the finish line before Achilles.
> > >>>>>>>>
> > >>>>>>>> The turtle will never cross the finish line but will always be ahead
> > >>>>>>>> of Achilles.
> > >>>>>>>
> > >>>>>>> Yes. True. It gets infinitely closer and closer to the finish line.
> > >>>>>>
> > >>>>>> That is not Zeno's Paradox. The tortoise gets a head start, at point
> > >>>>>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
> > >>>>>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
> > >>>>>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
> > >>>>>> forth.
> > >>>>>>
> > >>>>>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
> > >>>>>> [A3], ... get smaller and smaller, since the time it takes Achilles to
> > >>>>>> run from the start to A[1] equals the time it takes the slower tortoise
> > >>>>>> to run from A[1] to A[2], and so on.
> > >>>>>>
> > >>>>>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
> > >>>>>> always behind A[n+1] (the tortoise's position), even as n approaches
> > >>>>>> infinity. So Achilles can never beat the tortoise, right? But, as long
> > >>>>>> as the head start isn't _too_ large, in real life, Achilles passes the
> > >>>>>> tortoise and wins, just as you'd expect. So what's wrong with this?
> > >>>>>>
> > >>>>>> As I said, just because there's an infinite limit, it doesn't mean the
> > >>>>>> limit is absolute. In this case, the total time passed also reaches a
> > >>>>>> limit (at n=infinity) but that time limit isn't infinite, so what
> > >>>>>> happens after the "limit" on time passes? As always, time marches on...
> > >>>>>> At that point Achilles passes the tortoise and remains ahead for the
> > >>>>>> rest of the race, and the infinite series no longer applies.
> > >>>>>>>
> > >>>>>>>
> > >>>>>>>>> Now, if Achilles tells the tortoise to f-off and just starts
> > >>>>>>>>> running, he
> > >>>>>>>>> will quickly pass the tortoise...
> > >>>>>>
> > >>>>>> In real life, yes, but in Zeno's Paradox, no.
> > >>>>>>>>>
> > >>>>>>>>> ;^)
> > >>>>>>>
> > >>>>>>
> > >>>>>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
> > >>>>>> (S1>S2), and the head start distance A1, how long does it take for
> > >>>>>> Achilles to pass the tortoise? :-)
> > >>>>>>
> > >>>>> I did some equations on this a while back:
> > >>>>>
> > >>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
> > >>>>>
> > >>>>> Here are my comments:
> > >>>>>
> > >>>>> Iirc, scale was speed:
> > >>>>> ____________________________
> > >>>>> [...]
> > >>>>> Ahhhh, now this is a direct formula:
> > >>>>>
> > >>>>> n = iteration count
> > >>>>> d = distance
> > >>>>> s = scale
> > >>>>>
> > >>>>> r_[n] = (d / s^n) * (s^n - (s-1)^n)
> > >>>>>
> > >>>>>
> > >>>>> just might work for finding the total distance
> > >>>>> traveled at a given iteration count of the following
> > >>>>> iterated equation:
> > >>>>>
> > >>>>> r_[n+1] = r_[n] + (d - r_[n]) / s
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> Here is the sequence for d = 10 and s = 4 using the
> > >>>>> iterative formula:
> > >>>>> __________________________________
> > >>>>> r_[0] = 0
> > >>>>> r_[1] = 0 + (10 - 0) / 4 = 2.5
> > >>>>> r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
> > >>>>> r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
> > >>>>> r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
> > >>>>> __________________________________
> > >>>>>
> > >>>>>
> > >>>>> And here is the sequence for d = 10 and s = 4 using
> > >>>>> the direct formula:
> > >>>>> __________________________________
> > >>>>> r_[0] = 10 / 1 * 0 = 0
> > >>>>> r_[1] = 10 / 4 * 1 = 2.5
> > >>>>> r_[2] = 10 / 16 * 7 = 4.375
> > >>>>> r_[3] = 10 / 64 * 37 = 5.78125
> > >>>>> r_[4] = 10 / 256 * 175 = 6.8359375
> > >>>>> __________________________________
> > >>>>>
> > >>>>>
> > >>>>> As you can see, they are identical!
> > >>>>>
> > >>>>> Humm...
> > >>>>> ____________________________
> > >>>>>
> > >>>>>
> > >>>>> Here is another post:
> > >>>>>
> > >>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
> > >>>>> ____________________________
> > >>>>> I think I found a way to find the handicap of a
> > >>>>> runner in an infinite race on a finite track...
> > >>>>>
> > >>>>> How about something like:
> > >>>>>
> > >>>>>
> > >>>>> Let:
> > >>>>>
> > >>>>> d = total distance in track
> > >>>>> s = scale, which relates to speed
> > >>>>> n = integer iteration count, which relates to time
> > >>>>> r_h = a runners starting handicap
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> Here is the iterative equation for finding the
> > >>>>> distance a runner is down the track that I posted
> > >>>>> up thread:
> > >>>>>
> > >>>>> r_[n + 1] = r_[n] + (d - r_[n]) / s
> > >>>>>
> > >>>>>
> > >>>>> The handicap of the runner is equal to r_[0]
> > >>>>> because n = 0 is the starting position of every
> > >>>>> runner.
> > >>>>>
> > >>>>> The goal is to find the handicap of a runner with
> > >>>>> a given distance, iteration count, total distance
> > >>>>> of the track, and a scale or speed. AFAICT, the
> > >>>>> following formula solves for the handicap of a
> > >>>>> runner using that information:
> > >>>>>
> > >>>>>
> > >>>>> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> Here is output of a racer using the iterative equation
> > >>>>> with the following attributes:
> > >>>>>
> > >>>>> d = 10
> > >>>>> s = 4
> > >>>>> r_h = 6.8
> > >>>>> _______________________________________
> > >>>>> r_[0] = 6.8
> > >>>>> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
> > >>>>> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
> > >>>>> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
> > >>>>> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
> > >>>>> _______________________________________
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> As we can see this runner has a head start of 6.8 out
> > >>>>> of 10. Also, in the third frame, the runner r_[2] has
> > >>>>> traveled 8.2 out of a possible 10.0.
> > >>>>>
> > >>>>> Given that information alone, we can plug it all into
> > >>>>> the formula for finding the handicap, and get:
> > >>>>>
> > >>>>>
> > >>>>> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> Bingo! We now know that the handicap for the runner
> > >>>>> is 6.8 at n = 0 by information reaped in a later moment
> > >>>>> in time when n = 2... Three frames later.
> > >>>>>
> > >>>>>
> > >>>>> Is this Kosher?!?!
> > >>>>>
> > >>>>>
> > >>>>>
> > >>>>> :^o
> > >>>>>
> > >>>>> ____________________________
> > >>>>
> > >>>> If you add zero to .999 repeating you still get .999 repeating.
> > >>>> Add the infinitely small and you get 1 instead.
> > >>> .999 repeating = 1.000 repeating anyway
> > >> Mitch, for that ".999... is add infinitesimal", just first
> > >> have it that "1 minus infinitesimal, is, .999..., lesser".
> > >
> > > .999 is lesser than one by the infinitely small not zero.
> > >
> > > Mitchell Raemsch
> > if you add an infinitesimal to 1 you get 1.000... repeating
> No. You get above 1 by the infinitely small.
>
> Mitchell Raemsch