Ross A. Finlayson used his keyboard to write :
> On Wednesday, June 1, 2022 at 2:10:24 PM UTC-7, Michael Moroney wrote:
>> On 6/1/2022 2:00 PM, mitchr...@gmail.com wrote:
>>> 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.
>> Who are "we", Mitch? You and Roy Masters?
>
> It really only works to define infinitesimals uniformly from zero through
> one, then those elements named "iota-values" are contiguous, that "EF is a
> function" and "EF(1) > EF(0)", but it's not really the usual attachment to
> division, where it's written according to the existence of the limit that
> 1/oo = 0.
>
> Also "iota-sums" and "iota-mutiples" are not the same.
>
> In something like "surreal numbers", there is something like "1/w".
>
> Given its rank or order, or "in omega", the infinitely many regions
> [0/w, 1/w], [1/w, 2/w], ..., are, like infinitely many regions and have
> trichotomy and are well-ordered and countably many and so on.
>
> https://en.wikipedia.org/wiki/Surreal_number
>
> Of course Conway's one of the most famous mathematicians of our time.

I think that the birthday idea rocks. No more confusion over redundant
representations.