The basic solution to the paradox is integral calculus. The normal statement of the paradox keeps using smaller and smaller time intervals. Integral calculus lets you take an infinite number of infinitely small areas and have it add up to a finite answer. (Which is scary at first, but you get used to it pretty quick.) Once you have that insight, the paradox goes away and the normal view of velocity versus time gets strengthened.

Like all paradoxes it is only a problem because when proposed there was a lack of mathematics knowledge. Lets look at the paradox in a different way:

How long does it take Achilles to reach the tortoise?

One way to solve this is to just add up all the time intervals for each step. So it takes Achiiles some time, t^1 to get to point A, t^2 to get to point B and so on. Our total time, T, is then:

T= t^1 + t^2 + t^3 .... for an infinite number of intervals.

So how do we get a finite number from adding an infinite number of positive values together? Without calculus we can't solve this and hence the paradox.

The reason this is a "paradox" is that the logic seems irrefutable although our common sense tells us otherwise. It isn't a true paradox because it isn't a logical contradiction but rather the reasoning seems to go against common sense.

To actually show why this isn't a true paradox involves understanding infinite series. We can build an infinite series out of the sequence 1, 1/2, 1/4, 1/8... Now every term of the sequence is positive. "Logically" adding all the terms would result in "infinity" as there are an infinite number of positive numbers added together.

It actually isn't obvious to a person not familiar with infinite series, why this "logic" isn't true.

The paradox is that you have one line of reasoning that shows that Achilles will never reach the Tortoise, and another line of reasoning that shows that Achilles will eventually reach the Tortoise.

If both lines of reasoning are correct, you get a contradiction. But it's not obvious where the mistake is, hence being called a paradox.

The resolution to this paradox is the realization that an infinite series can have a finite sum. That is, the first line of reasoning shows that it will take an infinite number of steps for Achilles to reach the Tortoise, but since each step gets shorter, it can be done in a finite amount of time.

Imagine that I gave you a task:

"Start counting up in integers (1, 2, 3, etc.). When you finish counting all the integers then you can have a slice of cake."

When are you going to have your slice of cake? You are never going to run out of integers since you can keep counting up for an infinite period of time, so in theory you should never reach the point where you can have the cake, right?

Now imagine that as you are counting the time between 1 and 2 takes a second, but you start speeding up so that the time from 2 to 3 is only a half second, from 3 to 4 a quarter of a second, etc. Conceptually this doesn't matter since we didn't really care about how quickly you were counting in the first example, as the issue of the integers being infinite was the real issue. But in this case somehow you "finish" and eat the cake?

This is the idea behind Zeno's Paradox. In order for the hare to pass the tortoise it must first reach the tortoise, and in order to do that it must reach half the distance between it and the tortoise. If it reaches the halfway point then it must next reach the new halfway point, and when it does that reach the new new halfway point, etc. In concept this cycle can be continued infinitely since distance is infinitely divisible and so there are equally an infinite number of steps to this task as there are integers in the previous examples. Yet in this case we know the hare will be able to pass the tortoise. Somehow an infinite series of tasks was completed in finite time.

Why should we assume their speed is different if they are both advancing one point at a time? I feel like I’m missing something.

We use the word paradox in a couple different ways. One is in logical contradictions, i. e. "This sentence is false." It is impossible to have the statement be either true or false, so it is a paradox. "A married bachelor" is another paradoxical statement, as you are definitionally bound by the word "bachelor" to be unmarried.

Another way we use paradoxes is in seemingly contradictory statement that do actually have an answer. Zeno's paradox falls in this category. Like you say, there is the definite answer that Achilles does catch the tortoise, the seemingly contradictory part is the description that you have to cover an infinite number of ever smaller but never 0 distances to go anywhere, so how can you move?

The answer being that the time it takes to move those smaller increments also decreases to 0 at exactly the same rate.

Not all paradoxes can be solved with mathematics.

This sentence is false. That's a paradox as well. It cannot be true nor false. No amount of mathematics can make it true or false, it is a logical impossibility.

But there remains the problem of whether, and if so how, the hare actually does move through an infinite number of states in reality, which is why you still see some people arguing that the paradox is unresolved, or at least that it requires a more elaborate solution than pointing out that an infinite geometric series can have a finite sum (which doesn't actually need calculus, and I think was known by some Greeks slightly less Ancient than Zeno).

This is not a paradox for 2 reasons:

1. You are assuming language is logically consistent, it is not.

2. You are assuming logically consistent systems are also complete, they are not.

Let's give some formulas for Achilles and the Tortoise, to see how the times when Achilles reaches a point where the Tortoise was previously at keeps shrinking and stays below a definite time (when their positions are tied) rather than getting big.

Say Achilles has position A(t) = t at time t and the Tortoise has position T(t) = 4 + t/100 at time t, so A(0) = 0 and T(0) = 4: at time 0, the tortoise is 4 units ahead (meters, feet, whatever).

Achilles reaches position 4 at time 4: A(4) = 4. And at that time T(4) = 4.04 > 4, so the tortoise is still ahead at time 4.

Achilles reaches position 4.04 at time 4.04: A(4.04) = 4.04. And at that time T(4.04) = 4.0404 > 4.04, so the tortoise is still ahead at time 4.04.

Achilles reaches position 4.0404 at time 4.0404: A(4.0404) = 4.0404. And at that time T(4.0404) = 4.040404 > 4.0404, so the tortoise is still ahead.

Achilles reaches position 4.040404 at time 4.040404: A(4.040404) = 4.040404. And at that time T(4.040404) = 4.04040404 > 4.040404, so the tortoise is still ahead.

Despite the tortoise always being ahead of Achilles at these times, notice the times we are working with are always remaining below 4.040404040404... < 4.05. So it's simply false that Achilles "never" catches the tortoise because such reasoning is just ignoring the actual passage of time by focusing on ever vanishingly small units of time. Once the time reaches 400/99 = 4.040404040404..., Achilles and the tortoise are at the same position: their positions are tied. And after this time Achilles gets ahead of the tortoise and remains ahead. Notice the time when Achilles and the tortoise are tied is the value of an infinite decimal 4.0404040404..., which can be thought of as a convergent infinite series 4 + .04 + .0004 + .000004 + ..., which is related to the answer by u/EquinoctialPie saying that the resolution of the paradox is the fact that infinite series can have finite values. One does not need other tools from calculus (like integrals), but perhaps the series can be viewed in different ways related to those other tools.

You can graph this: plot y = t and y = 4 + t/100. For very small positive t, we have t < 4 + t/100 (Achilles is behind the tortoise), but when t = 400/99 the two lines cross, and when t > 400/99 the first line is higher than the second line (Achilles is ahead of the tortoise).

In summary, there is no actual paradox if you pay attention to all times instead of getting fixated only on quite small times. The error here is analogous to people who mistakenly think an unending decimal like pi = 3.141592653589... is an "infinite number" because it has infinitely many digits: this confuses the number of digits with the numerical value of the decimal.

Besides the fact that one is a tortoise you mean?

Sure, but OP specifically says that they are advancing one step at a time, at the same time.

I'm not a math dude, so I can't confirm that step means something specific in math/coding, but I think step means, like, "feet touching grass"

I suppose I am assuming A, B, C…are equidistant.

Another cause/fix to it is to do with quantisation, and quantum physics.
This lead to a similar paradox called the UV catastrophe.

The premise was you could infinitely divide things, time and distance, so half of whatever measurement there is, but it turns out you can only get so far, and then it blends together in the Planck length and Planck-time.
It's not a case of just being the smallest thing currently measurable, it's more the limit on the granularity of the universe.

The UV catastrophe is what started quantum physics, the maths said that a black body (in simple words, something that radiates heat perfectly) would radiate energy at all different frequencies in differing levels. The maths divided these frequencies infinitely, and they all had at least some energy, at that would mean it would radiate infinite energy too.
It was figured out that you couldn't just divide everything infinitely, it got down to discrete quantised steps, eg mini packets or "quanta" in these levels.

The Planck length is the smallest possible division of space and distance, and you can't halve it, and the Planck time is the time it takes light to cover the Planck distance, eg the fastest possible thing covering the smallest possible distance, and it can't get smaller.

Tortoise has a head start, but moves slow.

Achilles will overtake and win despite starting behind.

The problem is an infinite series.

Achilles will run, for some amount of time, and eventually reach the tortoise's starting point. But the tortoise has also advanced during that time and Achilles is still behind.

The race continues a lesser amount of time and Achilles reaches the tortoise's last checkpoint, but again the tortoise has moved on.

EVERY time Achilles reaches where the tortoise was, the tortoise has moved a little bit farther. This will be true for infinite points.

How can Achilles reach the infinite number of checkpoints and ever pass the tortoise if the tortoise is always a little bit further away for infinite points?

An infinite series of mathematics walk into a bar.

The first orders a beer. The second orders half a beer. The next orders 1/4 of a beer. Next orders 1/8 of a beer.

The bartender cuts them off, pours 2 beers and tells them "you should really know your limits."

Ok. This is a good way to put it.

Achilles definitely catches the tortoise.

The way the story is told, they keep describing snapshots in time that are closer and closer to the time the tortoise is caught without telling you about that moment.

Like if he catches him at noon and you tell a story describing 11:59, then 11:59:30, then 11:59:45, then 11:59:52.5, etc.

You can certainly tell a story like that where you never get to noon. But we all know how time works. Eventually noon will come.

Zeno argues that motion is impossible because in order to get from one point to another, an object must first travel half the distance, then half the remaining distance, and so on, ad infinitum. Since there are an infinite number of distances to be traveled, the object can never actually reach its destination.

In reality, Achilles will eventually catch up to the tortoise, despite the paradox's argument that he cannot. Zeno's paradox is based on the assumption that an infinite number of smaller and smaller distances must be covered in order to reach a destination, but in reality, there is a smallest possible distance that can be traveled, such as the Planck length in physics.

Furthermore, the paradox assumes that time is infinitely divisible, which is also not true according to modern physics. When the actual physical laws are taken into account, the paradox can be resolved, and it becomes clear that Achilles can indeed catch the tortoise, given enough time.

Zeno's paradoxes continue to be interesting philosophical puzzles that raise questions about the nature of space, time, and motion, but their solutions lie in our understanding of modern science and mathematics, which provides a more accurate and realistic description of the world around us.

Outside of some tricks with language ("this statement is a lie"), you can't have actual paradoxes. By definition, a real paradox is impossible. However, you can get things that seem paradoxical. Normally, these situations involve reasoning that seems good, based on common assumptions, that leads to a result that contradicts what we know to be true.

So, we know that a racer can overtake another racer that has a headstart on him. Zeno's reasoning seems solid, but indicates that overtaking should be impossible. The common assumption that turns out to be wrong is the idea that space is infinitely divisible. We're in a simulation, and you can't have half-a-pixel, so you have to always move across the screen in discrete units.

I always thought the soloution was e will pass when his indiviual strides are longer than the tortoises distance traveled during the time it takes Achilles to make one stride

To complete a stride, Achilles foot would have to move to half way through his stride, and to do that it would need to move half way to that , and so on.
That just shifts the thing down.

The universe having some kind of finite resolution like the planck length would also resolve this.
But it's not necessary, for the reasons stated above.

Oh so you have a set that’s open on one end. That is really weird since you finished but there is no last number. It’s like the lamp version of this.

>You are assuming language is logically consistent, it is not.

You declaring something to be true (or not true) does not make it so.

Nor does the paradox (it is a paradox btw, you don't get to unilaterally define what a paradox is or is not and the above is definitely accepted as a valid paradox) depending ALL language being logically consistent, it is, in fact that language can express logically inconsistent statements that allows paradoxes.

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>You are assuming logically consistent systems are also complete, they are not.

This has literally nothing to do with the original statement OR the comment you are replying to.