onsdag 1 juni 2022 kl. 20:00:54 UTC+2 skrev mitchr...@gmail.com:
> On Wednesday, June 1, 2022 at 9:52:27 AM UTC-7, Ross A. Finlayson wrote:
> > On Wednesday, June 1, 2022 at 9:43:27 AM UTC-7, sergi o wrote:
> > > On 6/1/2022 11:00 AM, Ross A. Finlayson wrote:
> > > > On Tuesday, May 31, 2022 at 9:40:03 PM UTC-7, sergi o wrote:
> > > >> On 5/31/2022 3:15 PM, mitchr...@gmail.com wrote:
> > > >>> On Monday, May 30, 2022 at 9:54:09 PM UTC-7, zelos...@gmail.com wrote:
> > > >>>> fredag 27 maj 2022 kl. 19:38:33 UTC+2 skrev mitchr...@gmail.com:
> > > >>>>> 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.
> > > >>>> There are no infinitesimals in real numbers. The real numbers are archimedian. I have told you this
> > > >>>
> > > >>> How do you know they are more real than the Calculus fundamental infinitesimal?
> > > >>> .999 repeating is not the same quantity as the first integer.
> > > >>> Add zero to .999 repeating and you get .999 repeating.
> > > >>>
> > > >>> Mitchell Raemsch
> > > >> how do you know you actually have an infinitesimal ?
> > > >
> > > > Deduction: "continuous exists? could not be not infinitesimal".
> > > >
> > > > It's more that you know that you _don't_ have an infinitesimal, but,
> > > > that according to the existence of some analog process like the
> > > > procedure in time, that "effectively" that given any specific frequency
> > > > of otherwise finite events, there's another of not-necessarily finite,
> > > > "effectively", events. (That includes them.)
> > > >
> > > > Basically that time goes on forever and never stops.
> > > >
> > > > Or a mathematical model of same, ....
> > > >
> > > > Deduction, that's how. Deductive inference is what's seated under
> > > > inference, anyways. (This) ...after complementary terms, and
> > > > complementarity of course is of greatest grounds for deduction.
> > > >
> > > > "Infinite" is a qualia, if it's the numbers, not ours.
> > > >
> > > > "Atomism" is probably a most familiar theory for
> > > > "effectively, ..., infinitesimal atoms exist". Beyond that,
> > > > then, there's superstring theory, "atoms' infinitesimal
> > > > superstrings exist". That's about it, with atomic scale about
> > > > 25 orders of magnitude and superstring scale about 50,
> > > > orders of magnitude smaller than 1.0 meter.
> > > >
> > > > In theory, ....
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > you could be covered in infinitesimals, and not know it, they do itch though.
> > By the time we had formal real analysis after a theory of limits,
> > the other courses included atomism, particle/wave duality, ....
> >
> > Avogadro's number is a stoichiometric constant relating
> > abstractly indistinguishable atoms in count, to mass, kinetic.
> >
> > It's a finite number.
> >
> > Anyways if there are "finitesimals" or not, if not, then "infinitesimals".
> >
> > Here of course "finitesimals isn't a word", but, it just means smallest
> > quantities in some fixed-point arithmetic, for example, then that in
> > the unbounded, those are arbitrarily small.
> We know 1 divided by infinity is just as real as any other quantity.
> A fundamental creates an first quantity.

False, there is no "infinite" in real numbers and thus 1/infinity is not real either.