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I heard Zeno's "Achilles and the Tortoise" paradox is resolved using the limit concept of mathematics, i.e., the time cost in every sub-race is added up, which has a limit, not limitless, so Achilles can catch up the Tortoise finally.

That resolution is based on two assumptions:

  1. the catch-up time equals to the sum of a series
  2. the series is convergent and the sum of the series is a limited number

The second assumption can be strictly proved in math. But how to prove the first assumption?

Bernard
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William
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    It might help if you stated what you think the paradox consists of, so we can point out the series – Henry Jul 27 '20 at 09:08
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    I don't see how this is any kind of paradox--it's a linear equation to be solved. – Jared Jul 30 '20 at 15:10
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    Isn't it more or less a duplicate of https://math.stackexchange.com/q/3762872 ? – Paul Frost Jul 30 '20 at 15:22
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    This question is a duplicate. Please do your research before asking questions. – K.defaoite Jul 30 '20 at 15:23
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    It is a duplicate of a heavily upvoted post. It is hard to beliebe, but apparently paradoxes of the antique can still confuse people today (See the heated and pointless discussion) .The resolution of this "paradox" is so trivial that it does not even deserve to be called a paradox. – Peter Jul 30 '20 at 16:42
  • @K.defaoite I read the top answers to that question and found they do not answer my question. My question is not the same as that one. Physically, Achilles will catch up the Tortoise in t=d/(Va-Vt) and that value happens to the same as the sum of the series. But I do not think the elementary equation is obvious and does not need a math proof. – William Jul 30 '20 at 16:52
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    I don't think it really is a duplicate of that post, since that post is "Is my characterization right and if so how is my proof wrong". This is asking about the general idea of the paradox, not some specific formalization of it. – DanielV Jul 30 '20 at 17:23
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    If you really want to thoroughly investigate this, see the references I gave in my answer to Achilles and the tortoise paradox? – Dave L. Renfro Jul 30 '20 at 18:55
  • @Dave L. Renfro Your references are very abundant. I hope I will have time to read them all in future. The third answer to that question is also instructive, but people concerning to this paradox seem to like the kind of explanation as the first answer to that question. I do not know why. – William Jul 31 '20 at 13:32
  • "But how to prove the first assumption?" What do you mean? That's in the statement of the paradox. What's to prove? – fleablood Aug 05 '20 at 05:42
  • "But I do not think the elementary equation is obvious and does not need a math proof." Really? If $d_k = t_k - a_k$ and $time_k = \frac {d_k}{v_a}$ and $a_{k+1} = a_k + t_kv_a$ and $t_{k+1}=t_k + t_kv_t$. You don't see that the total amount of time is $T=\sum_{k=1}^\infty time_k=\sum \frac {d_k}{v_a}$ is not clear and needs a "proof"? – fleablood Aug 05 '20 at 05:49
  • I updated my answer in response to a comment. – it's a hire car baby Aug 05 '20 at 06:36
  • @fleablood T is a definite number you assume it exists, $\sum_{k=1}^\infty time_k$ is a dynamic process. You can not say they are equal in value without proof. It may not even be natural to consider the "=" as a synonyms of "defined by". – William Aug 05 '20 at 07:22
  • @fleablood The elementary resolution to the paradox is: "supposing Achilles can catch up Tortoise and the catch-up time is T, then T satisfies VaT=VtT+d, that equation does have a solution T=d/(Va-Vt), so Achilles can indeed catch up Tortoise in limited time". The proof seems to have a loop in it. – William Aug 05 '20 at 07:43
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    @William well, I'd call that an avoidance of the paradox rather than a resolution. Of course faster Achilles will catch up with the tortoise given a head start. And we can calculate how long it takes (but why? who cares?) but that doesn't address the paradox: which is that there are an infinite number of times where achiles must reach the points where the tortoise was when achiles reached the previous points. The total time can be set up as an infinite sum. If we accept infinite points in time and infinite sums as doable there is no paradox. But infinite tasks in finite times was the paradox – fleablood Aug 05 '20 at 15:37
  • @William 'The proof seems to have a loop in it.' No: It is an assumption of the model of the physical situation that Achilles will catch up with the Tortoise. One then uses some elementary mathematics to calculate exactly when Achilles catches up with the Tortoise. – dwolfeu Aug 06 '20 at 11:59
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    Also related: https://math.stackexchange.com/questions/3915067/can-someone-explain-the-relation-between-achilles-chasing-turtle-paradox-and-m and https://math.stackexchange.com/questions/1217814/interpretation-of-zenos-paradox-in-gelfands-algebra-text and https://math.stackexchange.com/questions/3533835/sum-of-infinite-geometric-progression-paradox-of-zeno and https://math.stackexchange.com/questions/335560/is-1-divided-by-3-equal-to-0-333 and https://math.stackexchange.com/questions/3870175/zeno-paradox-induction-proof and probably more. – Gerry Myerson Aug 26 '21 at 02:22
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    If you really want to address any of Zeno's well-known motion paradoxes then you are asking in the wrong forum. The paradox is philosophical, not mathematical. The mathematical solutions all assume the validity of one or another particular model of reality; the philosophical argument disputes the validity of the model. Look a little further into Zeno's work and you'll find he not only "proved" Achilles cannot catch the tortoise, he "proved" Achilles cannot reach an unmoving finish line, in fact Achilles cannot even get off the starting blocks. – David K Aug 26 '21 at 03:09
  • @DavidK Absolutely right ! I think nothing has to be added to this comment. However, a lot of people seem to have another opinion. I already mentioned it and mention it again : This "paradox" can so trivially be resolved that it does not even deserve to be called a paradox. – Peter Aug 26 '21 at 06:53
  • @DavidK "The mathematical solutions all assume the validity of one or another particular model of reality ". In this case, can you explain the model of reality that is assumed to be valid? – William Aug 26 '21 at 10:00
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    I said "one or another particular model," so there is not necessarily the model that all solutions use. But some features of many of these models are that time since the start can be measured by a real number $t,$ Achilles experiences time $t$ for every real number $t>0,$ and at time $t$ Achilles has traveled a distance $vt$ where $v$ is Achilles' constant speed. That's a lot of assumptions. – David K Aug 26 '21 at 11:21
  • By the way, I myself fully buy into the typical mathematical models for all practical purposes, and I am not a good person to explain Zeno's objections. Ask a philosopher who specializes in this sort of thing instead. You're not likely to find that person on math.stackexchange. – David K Aug 26 '21 at 11:49
  • @DavidK I'm interested in the general concept of "model". Is there accurate definition of this word? Is there a math branch specialized on "model". I hear of "Model Theory" in computer science but do not know if it has the same meaning. – William Aug 26 '21 at 15:19

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I think the assumption follows from the analysis of the motion before the catch. We see that it splits into stages.

Stage 1: Achilles is running towards the Tortoise starting point $A_1$, the Tortoise is running forward. At the moment when Achilles achieves $A_1$, Stage 1 ends and begins Stage 2. At this moment the Tortoise is at point $A_2$.

Stage 2: Achilles is running towards $A_2$, the Tortoise is running forward. At the moment when Achilles achieves $A_2$, Stage 2 ends and begins Stage 3. At this moment the Tortoise is at point $A_3$.

And so forth...

The construction of the stages show that each of them happens before the catch. This is a surprising view on a usual motion, but it sounds reasonable. So, we have that catch-up time $T_C $ is at least the sum $T_\infty=\sum T_n $ of the series of the durations of the stages $T_n$.

To show the assumption it remains to show that a strict inequality $T_C>T_\infty$ is impossible. Indeed, suppose to the contrary that after time $T_\infty$ Achilles is still behind the Tortoise. This means that a distance $d_\infty$ between them is non-zero. Let $d_0$ be the initial distance between Achilles and the Tortoise. Since Achilles is faster than the Tortoise, the $q=v_A/v_T$ of their velocities is bigger than $1$ and the distance between them decreases. But induction we can easily show that the distance $d_n$ between Achilles and the Tortoise after Stage $n$ is $d_0/q^n$. Since $q>1$, there exists $n$ such that $d_0/q^n<d_\infty$. But this is impossible, because the distance between Achilles and the Tortoise decreases and the end of Stage $n$ happens before $T_\infty$.

Alex Ravsky
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  • Your logic sounds solid, but use indirect proof several times, which is not very intuitive. – William Jul 31 '20 at 04:59
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    @William Zeno’s aporiae are very subtle. It is easy directly calculate the catch-up time and the sum of the series and check that they are equal. But I proposed and analyzed a motion model, making explicit the used assumptions (in order to discuss them, to agree with them as with obvious and natural, or to disagree). This is like the ancient Greeks did, for instance, Euclides in “Elements”. – Alex Ravsky Jul 31 '20 at 05:41
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    I have never seen such a explanation as yours before. Your motion model and proof thereafter should be original. – William Jul 31 '20 at 13:37
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I'll add an answer that will sound cheeky but, I insist, hits the point:

You're asking for a reason why "the catch-up time equals the sum of a series".

Well:

A) Time is measured in real numbers.

B) Real numbers are well-defined in mathematics: They are, in one disguise or another, "certain sums of series". That's why we define real numbers the way we do, most commonly by Cauchy sequences or Dedekind cuts, which both can be translated to "this real number is (among many other things) this sum of a series".

To philosophers and physicists, the exact mathematical definitions of the real numbers might seem technical; but we cherish and laud them as an important breakthrough, because they are consistent and make the paradox disappear. Note the two ways to compute the time that Achilles passes the Tortoise in dwolfeu's answer: They give a consistent result based on that consistent theory of real numbers. It's the very definition of $\mathbb R$ which makes the results of the two computations, one seemingly purely algebraic, the other seemingly relying on a limit from calculus, identical.

One can, of course, doubt A). One could say, nobody has ever measured a time interval of $12\pi$ seconds or $\sqrt {15}$ hours (or: maybe sometimes it should come out as $-4i$ days). Fair enough (although then I would add, has anyone ever truly and beyond all doubts measured a time interval of $\frac{15}{17}$ minutes either?). However, in all actual experiments, all "Achilleses" have actually overtaken all "Tortoises" at times which to the precision of measurement possible agreed with what the mathematical model of real numbers gave out. And, honestly, what more could one want?

Torsten Schoeneberg
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  • From your A and B, it seems to say "Achilles can catch up the Tortoise in a sum of series. The sum of series is defined by a real number t. So we say Achilles can catch up the Tortoise in t." However, my question is why Achilles CAN catch up the Tortoise in a sum of series. – William Oct 09 '21 at 10:57
  • I don't understand. What do you doubt: That we measure time in real numbers? -- Or, let me ask: Is there any model of movement which you would not doubt? If Achilles and his brother start 100m apart, and move straight towards each other with 1 m/s, do you doubt that they CAN meet in 50 seconds? Are there certain kinds of numbers and operations (finite sums, differences, quotients?) which you think are allowed in such questions, and others (infinite sums?) which are not? Why that distinction? If you say none is allowed, i.e. you question why anything can ever happen at all, I cannot help ... – Torsten Schoeneberg Oct 09 '21 at 15:32
  • I do not doubt these physical phenomenons can happen. I just doubt the math operations like sum of series or real numbers that claim they can prove the physical phenomenon can happen. – William Oct 09 '21 at 16:40
  • And if you are about to prove those physical phenomenon can happen in math, you must get rid of the presupposition that those physical phenomenon can happen in the first place, otherwise, you would be in a situation of loop-proof. – William Oct 09 '21 at 17:02
  • Well, one can have the view that mathematical operations cannot prove anything about physical phenomena. But then why do mathematics, or ask questions on a math site, at all? Again let me ask: If Achilles A and his brother B start 100m apart from each other, and then walk towards each other both with velocity 1 m/s -- do you believe a) that they will physically meet after 50 seconds b) that the mathematical calculation we do to come up with those 50 seconds has any meaning, or does it just accidentally match the outcome of the physical experiment? – Torsten Schoeneberg Oct 10 '21 at 01:59
  • I'll put it this way: The theory of real numbers (including calculus and limits of infinite series) does not, strictly speaking, "prove" anything about physical phenomena. However, if we use it to model what in Zeno's view (who did not know calculus) leads to some undefined never-ending process, it actually gives out results which in all cases known match both the measurement of physical experiments and, if available, more elementary ways of calculating meeting points (here, simple algebra). – Torsten Schoeneberg Oct 10 '21 at 02:05
  • I think the key point of my confusion originates from the obscure term "model". There must be a math branch explaining the term in a systematic way. Do you know what it is? – William Oct 10 '21 at 03:01
  • There is a branch of logic called "model theory", but that is not the sense in which I use the word here (and I think neither does @DavidK in comments with which I agree). I use it loosely as a philosophical term. And au contraire: There cannot be a a math branch formalizing that concept, because if it were a mathematical concept, it would still fail to bridge the gap to "real life", and you could still say "but that math proves nothing about physical phenomena". (Cf. my answer https://math.stackexchange.com/a/4228516/96384, addressing a similar issue in a very different context.) – Torsten Schoeneberg Oct 10 '21 at 04:51
  • I read your answer to that question, which is helpful. I will do some research on model theory and see if I can get some new findings. Thank you, @TorstenSchoeneberg – William Oct 10 '21 at 06:05
  • You can do that, but I repeat, I did not use the word "model" in the sense of model theory, and I expect at best vague analogies between my use and the use there. – Torsten Schoeneberg Oct 12 '21 at 19:16
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Imagine you are set the following homework problem:

Problem. Achilles and the Tortoise are moving along the same straight line in the same direction. Achilles is moving at 10 m/s and the Tortoise 0.1 m/s. At $t=0$ the Tortoise is 100 m ahead of Achilles. Let $T$ be the catch-up time, i.e. the time at which Achilles and the Tortoise meet. What is the value of $T$?

Solution 1. At $t=T$, Achilles will have travelled 100 m more than the Tortoise. Distance = speed $\times$ time, so we arrive the equation $10 \cdot T = 0.1 \cdot T + 100$. We rearrange to get $T=\frac{100}{9.9}\,\text{s}$.

Solution 2. Let's partition $T$ into parts and add them up. Let $T_1$ be the time at which Achilles has reached the Tortoise's starting point (so $T_1=10$) and in general let $T_{n+1}$ be the time elapsed between Achilles moving from his position at $t = \sum_{i=1}^n T_i$ to the Tortoise's position at $t = \sum_{i=1}^n T_i$. Then $T_{n+1} = \frac{0.1\cdot T_n}{10}=10^{-2} \cdot T_n$ and a straightforward induction shows that $T_n = 10^{3-2n}$. We can use the formula for a geometric series to add up the infinite sum: \begin{equation*} T = \sum_{i=1}^\infty T_i = \sum_{i=1}^\infty 10^{3-2i} = 10^3\cdot\lim_{n\to\infty} \sum_{i=1}^n\left(10^{-2}\right)^i = 10^3 \cdot \left(\frac{1}{1-10^{-2}}-1\right) = \frac{100}{9.9} \,\text{s} \end{equation*}

Remarks. I chose particular numbers (100m, 10 m/s, 0.1 m/s) in order to make the calculations easier and more readily understandable, but the calculations straightforwardly generalise to prove both assumptions given in the OP's question. Solution 2 is of course Zeno's paradox but without any mention of the word 'paradox'. It is a legitimate way to partition $T$, in the same way that it is legitimate to partition unity and write $1=0.\dot{9} = \sum_{i=1}^\infty \frac{9}{10^i}$. Lastly, note that Zeno's paradox talks about Achilles always being behind the Tortoise. The word 'always' is a temporal adverb and in the context of the paradox time is $T_i$, which vanishes. There is no "outside" time.

dwolfeu
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  • "Lastly, note that Zeno's paradox talks about Achilles always being behind the Tortoise. " I always took that "always" to mean at every iteration of calculations which we iterate and iterate in distinct steps that every iteration will always have the achilles behind the tortoise. ... This would be the same as saying "I always get seasick when I'm on a boat". That has nothing to do with noting that every time measurable by a clock I am seasick, just each "time" I'm on a boat I get sick. .... It's a language colloquialism. – fleablood Aug 05 '20 at 05:56
  • A fair point, although the reasoning appears to be a paradox only if 'always' is taken in the temporal sense, since then it seems that Achilles can never reach the Tortoise for any amount of time (which of course he can). If 'always' means 'for every $n$' then there's no paradox. The crux is that the time intervals $T_i$ vanish and sum to a finite amount. – dwolfeu Aug 05 '20 at 06:08
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    I think the apparent paradox is you have task1 that must be completed and when completed leads to task 2 and you have an infinite number of these tasks which must be completed one after another and there is no last task. Intuitively such a situation seems like it can not be capable of ending. Or if it does end it does so be doing one last task (which can not exist)....Secondarily, if every task takes a finite amount of time intuitively doing an infinite number of them should take infinite time. (the second is resolved by infinite sums. the first by....well, thinking about it. – fleablood Aug 05 '20 at 15:56
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Short answer: When two procedures used to solve a problem lead to different results (a paradox), the solution to the paradox is to show why one of the procedures is wrong. It is NOT showing why the wrong procedure must, would, or should lead to the same result. A complete explanation, and the solution, are given below.

What Does Solving a Paradox Mean?

As mentioned at the beginning of my article (at: https://bit.ly/2IM76rF ), a paradox proposes the existence of two different results as a solutions for the same problem. These results are inconsistent with each other, depending on which procedure is used. Only one can be correct. As Brown and Moorcroft suggest, we are not looking for a mathematical demonstration that Achilles reaches the tortoise. Assuming they are both running in the same direction, we know he will. We can calculate the exact time, given the distance between the two and the two speeds, using a simple formula:

t = distance/(difference in velocity)

Instead, explaining or invalidating a paradox is to show a fault in the paradox formulation, or the proposed solving procedure, so that we can exclude this procedure and demonstrate that there is only one result for the original problem. The solution of a paradox is the answer to the question: “How does the paradox formulation misrepresent reality or logic?” That is, we need to show why the proposed method is conceptually wrong. Solving a paradox, invalidates the formulation of a problem proposed by the author of the paradox and leaves us with only valid procedures for solving the original problem.

Why were the previous proposed solutions for the paradox not satisfactory?

Most, if not all, the proposed solutions to Zeno’s paradox assume that Zeno’s proposed procedure is correct. The procedure seems to be logical when it is first introduced to us, but we will see that the procedure proposed by Zeno is conceptually incorrect. The authors then used a procedure similar to Zeno’s faulty procedure to reach the expected correct result for the original problem.

For example, the simple way mentioned earlier to solve the problem (not the paradox) can be examined using a spreadsheet. Given the assumptions in the diagram below, the time for Achilles to reach the tortoise is 5 seconds:

39 m / 7.8 m/s = 5 s

A simple mathematical solution for the problem

If we calculate a sum of an infinite series, as several mathematicians have suggested, we obtain the same result: Achilles will reach the tortoise. By applying any legitimate mathematical solution to the problem, we state, in another way that: We can prove that Achilles reaches the tortoise.

What is the problem then?

We know that Zeno is wrong: The fact that his procedure never ends, does not imply that Achilles will never reach the tortoise. We can prove this mathematically in many ways. Intuitively we all agree with the mathematicians. However, when mentally following the proposed repetitive procedure, the paradox puzzles our mind.

What are the facts?

  1. Asserting that Zeno’s procedure never ends is correct, as we can prove it by writing a recursive computer program that follows Zeno’s steps. The program will never end, and never will provide us with the expected result, because the condition for the end of the recursion process (Achilles reaches the tortoise) would never occur.
  2. Asserting that Achilles never reaches the tortoise is wrong, as we can prove that he does, by using several mathematical procedures.

Thus, we must conclude that Zeno’s procedure to solve the problem, is not correct.

The question is then: Why is Zeno’s procedure wrong? The key word in the previous assertions is “never”. Never implies time and the problem must be considered in the context of space and time.

The Explanation of Zeno's Paradox:

Zeno's proposition invites the solver to do a series of steps each time changing system of reference: STEP 1: The starting system of reference: The point where Achilles starts the race and the tortoise is well ahead, STEP 2: After a while, we are then asked to use a new system of reference: The point where Achilles reached and where the tortoise initially started, with the the tortoise now a bit further ahead, STEP 3: Then again we are asked to use, recursively, a new system of reference with the new starting point for Achilles and with the tortoise still further ahead, with every step we are asked to freeze the process and then continue by re-creating and examining the original problem using a different system of reference.

Today we know more about the relative motion of two bodies. Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions.

NOTE: The concept of a system or reference, or frame of reference, in elementary physics is founded on Einstein's Special Theory of Relativity, First Postulate: All velocities are measured relative to some frame of reference.

After Zeno's proposed first step, or first change of system of reference, the problem, as presented in the second step, is exactly the same as the original, the only change being a difference in "scale". No progress was achieved in solving the problem. Changing system of reference essentially restarts the problem-solving procedure. This realization implies that the problem is never going to reach a conclusion as the step by step procedure is reiterated. If the system of reference is changed at every step, our working spacetime shrinks with every step, the solution becomes elusive and the tortoise becomes apparently unreachable. Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution.

Our Solution (Why Zeno's Formulation Is Incorrect)

Our solution of Zeno's paradox can be summarized by the following statement:

"Zeno proposes observing the race only up to a certain point, using a system of reference, and then he asks us to stop and restart observing the race using a different system of reference. This implies that the problem is now equivalent to the original and necessarily implies that the proposed procedure for solving the problem will never end."

That is, we cannot change system of reference in the middle of a problem involving velocity, space and time, whether the frame of reference is openly stated, or implied.

As an analogy, you cannot solve a problem involving measurements by using English Imperial measures at the start of calculations and then switch to metric measures (without proper conversions) in the middle of calculations.

Zeno's trick works, and puzzles our mind, because we are used to assume one frame of reference when solving this type of problems.

An example

The following is not a “solution” of the paradox, but an example showing the difference it makes, when we solve the problem without changing the system of reference. In this example, the problem is formulated as closely as possible to Zeno’s formulation.

Zeno would agree that Achilles makes longer steps than the tortoise. Let’s assume that one Achilles-step is about 20 tortoise-steps long, and let’s also assume that both Achilles and the tortoise make the same number of steps in the same amount of time. For example, two steps per second (the exact amount doesn’t really matter). If the tortoise starts the race 20 Achilles-steps ahead of him, then after 20 steps Achilles reaches where the tortoise was (See diagram below: Tortoise starting point).

No change of system of reference

In the meantime, the tortoise has made 20 of her steps, and she is now one full Achilles-step ahead of him. We have not changed our system of reference. We referred to both starting points. These did not move relatively to each other. We could choose any fixed ground point. To please Zeno, let’s continue by referring to the tortoise starting point, where Achilles currently is. When both runners make one more step, step 21, the tortoise will have moved by one of her steps and she will still be ahead of Achilles by that one tortoise-step. Achilles is now one Achilles-step ahead of the tortoise starting point. Now, let’s continue, without changing the system of reference. This is the key point. We do not redefine the problem and use the current positions of the runners as new starting points, as Zeno proposes, but we refer to the information about the race we have already accumulated in our knowledge base. Achilles then completes his 22nd step, and he is two Achilles-steps ahead of the tortoise starting point. The tortoise will have completed her 22nd tortoise-step from her starting point. Hence the tortoise is now behind Achilles by 18 tortoise-steps. Thus, if we do not change the system of reference, the paradox does not appear.

ggori
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    When linking to your own site or content (or content that you are affiliated with), you must disclose your affiliation in the answer in order for it not to be considered spam. Having the same text in your username as the URL or mentioning it in your profile is not considered sufficient disclosure under Stack Exchange policy. – cigien Aug 26 '21 at 02:16
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    Full disclosure: The article linked is on Medium. It was written by me (Giuseppe Gori) and it includes the same explanation, with an additional historical context. – ggori Aug 26 '21 at 02:57
  • @gg70 This is a good explanation. But, first, the assertion " Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions." is not very convincing. Second, you use a fixed step(time interval) to consider every result in the middle, which seems to say "I close my eyes and calculate the distances using elementary algebra in my mind in the interval, then when I open my eyes, Achilles and the tortoise are right at my calculated positions. My elementary algebra says Achilles can be ahead of the tortoise... – William Aug 28 '21 at 15:31
  • @gg70 so the possibility Achilles can not catch up the tortoise does not exist." Dare you open your eyes in the unlimited instances in the interval to draw the conclusion? – William Aug 28 '21 at 15:36
  • Reply to William: You say: first, the assertion " Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions." is not very convincing. REPLY: See Einstein, Special Theory of Relativity, Frames of Reference. The notion is now part of elementary physics. – ggori Aug 29 '21 at 23:32
  • William, with regards to the second part of your comment, you are right. Asserting that Achilles never reaches the tortoise is wrong, as we can prove that he does, by using several mathematical procedures. I added this assertion to my explanation. Thanks. – ggori Aug 30 '21 at 16:13
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Both Achilles and the tortoise pass through a continuum of points. Over any interval, there are infinitely points that each must pass through. Each point is of zero length, and so it takes no time to pass across any single point.

So, yes, Achilles must pass through the half-way point then half-half-way point, etc. and infinitely many such defined points, and this is not a problem.

Doug M
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