Zeno's Achilles & Tortoise - Where exactly is the proof wrong?

(For those who don't know what this paradox is see Wikipedia or the Stanford Encyclopedia of Philosophy.)

Let us define $a_i$ and $b_i$ recursively
$$ a_0 = 0\\ b_0 = 1\\ a_i = a_{i-1} + (b_{i-1} - a_{i-1})\\ b_i = b_{i-1} + (b_{i-1} - a_{i-1})/2 $$

It is easy to prove that $b_i>a_i\ \forall i$ using induction.

Thus while $|b_i-a_i|$ tends to $0$ , we will never have $a_i>b_i$.

We can now just replace $a_0$ as Achilles start position and $b_0$ as Tortoise start position. And then subsequent positions of Achilles is given by $a_i$s (Achilles new position is = Tortoise old position, which is the $1^{st}$ recursion). And Tortoise is assumed to move at half the speed of Achilles. Tortoise positions are represented by $b_i$s. (So, new position of Tortoise = Old Position + 1/2 the distance traveled by Achilles, which is the $2^{nd}$ recursion.)

Given, we have proven $b_i>a_i\ \forall i$, thus I claim Achilles will always be behind Tortoise (He will come closer and closer but will never overtake).

Obviously, I'm wrong but exactly where / which step of the proof above? (Please provide the exact mathematical step/argument where I went wrong.)

Some further discussion: Basis the responses I got (which I'm unable to find fully convincing - and it maybe just me that I don't understand them well enough) I would like to add - In my opinion, the way I have defined $a_i$ and $b_i$ it is just a subset of positions that Achilles and Tortoise can take. In that subset what I have proved is correct i.e. Achilles cannot overtake Tortoise. But just in that subset <- And I think this is the key

Note that my $a_i$ and $b_i$ are all rational. I can embed infinite rationals between any 2 points on the real line. I think fundamentally the error in my proof is that I use induction on continuous variables. I'm not formally trained to express that mathematically in a precise way - Hence this question.

My question is not to challenge/discuss that Achilles will overtake or not etc or to come-up with another proof - My precise question is where exactly is my proof wrong.

Thanks

The issue is as follows. You have constructed an infinite sequence of times, at all of which Achilles is behind the tortoise. However, that doesn't mean that Achilles will always be behind the tortoise, because the set of times you have constructed is bounded. Suppose Achilles has unit speed. Then they reach positions $a_1$ and $b_1$ at time $1$, $a_2$ and $b_2$ at time $3/2$, $a_3$ and $b_3$ at time $7/4$, and so on. It is easy to verify that all these times are less than $2$, so your argument only implies the tortoise is ahead for $t<2$. (Indeed, $t=2$ is exactly when Achilles overtakes the tortoise.)

Summary

Your proof is completely correct, there's no mistake in it. The mistake lies in your interpretation of the result that you prove.

Mistake

Before I pinpoint your mistake, note that

$$b_n<2\:\:\forall \:n\in\mathbb N\quad \rm and \quad a_n<2\:\:\forall \:n\in\mathbb N$$

which immediately implies that whatever you're going to conclude from your proof holds true only for the time interval where the displacement of Achilles and the tortoise, both, is less than $2$ units. After that, your series cannot provide us any information on how the distance between them will change.

Now you correctly concluded that $a_n<b_n \:\:\forall \:n\in\mathbb N$, but this only holds true for $a,b<2$. And now if we translate this mathematical argument to our paradox, we see that our proof states that Achilles will stay behind the tortoise as long as both of them haven't reached the $2$ unit mark. As they slowly get closer and closer to the $2$ unit mark, the displacement between them will start getting smaller and smaller, until they reach the $2$ unit mark. At this point, our series formulation is of no use, since $a=b=2$ is outside the "domain" of our series. And physically we know that it is at this point ($2$ unit mark) that Achilles will overtake the tortoise.

So, all in all, the equations just told you that Achilles will stay behind the tortoise up till the $2$ unit mark. This conclusion, as we know, is completely true and matches with the physical reality that we expected.

Conclusion

Thus, neither your mathematical formulation, nor what it "really" predicts is at fault here. In fact, nothing is fallacious at all because the math agrees with the reality. You were just drawing the wrong conclusions.

You write "Given, we have proved $b_i > a_i, \forall i$ ,thus I claim Achilles will always be behind Tortoise (He will come closer and closer but will never overtake)." This contains two sentences. The first sentence may or may not be false, depending on your meaning of "always". The parenthetical sentence is unambiguously false.

You have proven that for all $i \in \{0,1,\dots\}$, $b_i > a_i$. You have not related $i$ to time. You have not, in fact, incorporated time in your model at all. Thus, the only sense of "always" leading to a valid first sense is "for all nonnegative $i$".

"He will come closer and closer but will never overtake." cannot be concluded from "$i \in \{0,1,\dots\}$, $b_i > a_i$". All you can say is, for the times corresponding to nonnegative values of $i$, he will come closer and closer and not overtake. Your derivation is completely mute to times not corresponding to nonnegative $i$.

Your argument sees a sequence of snapshots of Achilles successively approaching the Tortoise's position, but the times at which Achilles passes the Tortoise and subsequently leads the Tortoise are not visible. In fact, the data your argument uses cannot falsify the following: In fact, Achilles is moving vastly faster than expected in each time interval bounded by the instants modelled by the pair of indices $i,i+1$ for $i \geq 0$ -- starting from the position specified at the time corresponding to the index $i$, he runs forward, passing the Tortoise by 100 meters, then turns around, runs back to the position specified at the time corresponding to the index $i+1$, then turns around to face in the forward direction, completing the turn at the time corresponding to the index $i+1$.

Although each index corresponds to a time, there is nothing in your argument indicating that the set of times includes the time when Achilles passes the Tortoise, or any time afterwards. In short, the argument speaks to a specific set of times, but does not apply to all times.

Since you did not, let me try to map your $i$ variable to time.

Let's assume that both Achilles and the Tortoise have constant speed (this is important), and Achilles speed is 1 m/s. Thus, from your definition of $a_i$ and $b_i$, is easy to see that Tortoise speed must be 0.5 m/s.

You defined $a_i$ and $b_i$ as recursive functions, but it is also possible to define them as a continuous, real valued functions. The natural extension to your definition is:

$$ a_i = 2 - 2^{(1 - i)}\\ b_i = 2 - 2^{-i} $$

Since Achilles speed is 1 m/s, we have that the time is:

$$ t(i) = \frac{a_i}{1 \text{m/s}} = 2 - 2^{(1 - i)} $$

We can already see the problem, which is $t: \mathbb{R} \rightarrow (-\infty, 2)$, i.e. $t(i) < 2\ \forall i \in \mathbb{R}$. This means that, no matter the value of $i$, it can never describe a world where $t \ge 2$. This becomes clear if we graph $a$, $b$ and $i$ regarding to position and time:

There are some points to consider in this graph:

• The closer Achilles and the Tortoise gets to the crossing point, at 2 m, your defined $i$ variable quickly tends to infinity. In fact, $\lim_{i\rightarrow \infty} a_i = 2$, which is the crossing point.
• That is to say $i$ curve never crosses the $t=2$ line.
• The inverse function $i(t)$ is not defined in $\mathbb{R}$ for $t \ge 2$, as you can see: $$ i(t) = \log_2 \left(\frac{2}{2-t}\right) $$

So, this is where I disagree with your affirmation "Achilles will always be behind Tortoise": it may be true to say "Achilles will be behind Tortoise for all values of $i$", but $i$ can not describe the whole interval of physical time if the speeds are constant. Instead, $i$ is simply not defined for times where Achilles has overtook the tortoise.

The step where your proof goes wrong is between

Given, we have proved $b_i > a_i$ ∀𝑖

and

thus I claim Achilles will always be behind Tortoise (He will come closer and closer but will never overtake)

The variable $i$ is not time. What you have shown is that, as Achilles overtakes the Tortoise, there is an infinite set of moments where Achilles is still behind the Tortoise, but by smaller and smaller amounts. But you already knew that.

The problem here is that you are trying to find your mistake in the mathematics, but it is actually in how you are interpreting the math as a model of the world. You have proven that something is true for all $i$, but $i$ isn't even meaningful to Achilles and the Tortoise. Time is.

After all, it is also true that there is no $i$ for which Achilles is exactly 1/3 behind the Tortoise. Does that say that Achilles is never exactly that distance away?

It is wrong to say that the problem is with using induction on continuous variables. This is ordinary induction on the discrete variable $i$. The problem is all in the interpretation of your result.

I realize this answer is similar to FakeMod's, but perhaps the perspective is different enough to help.