What exactly is the paradox in Zeno's paradox?

Solution 1:

It's only a paradox if you assume that the sum of (countable) infinitely many numbers cannot be finite. But modern mathematics has no problem with infinite sums that yield finite results - in the case of Zeno's paradox, the sum in question is $$ \sum_{k=1}^\infty 2^{-k} = 1 \text{.} $$

Not everything that is called a paradox is actually a logical inconsistency. Quite often, things only seem inconsistent because we inadvertedly make an additional assumption, which turns out to be wrong. In the case of Zeno's paradox, that is the assumption that infinite sums cannot yield finite results.

Solution 2:

The paradox is that you need to do infinite "actions" to get to the turtle, therefore you never get to the turtle because humans can't do infinite actions in a finite amount of time.

Of course it's not a true paradox, it's well explained by "you scale time as you scale space therefore you have a finite number". Keep in mind we were in ancient Greece, things like "convergent series" were far from being defined.

Solution 3:

This is* a (pseudo)paradox of infinite divisibility. Some people even use similar arguments today as a rationale for finitism.

When we look at the whole, the whole story is eminently plausible. And with today's knowledge, we can even sketch the continuum over which the story happens, mark where the events in Zeno's story happen, and add up all of the durations to see that the result is precisely the time it takes for Achilles to overtake the tortoise.

But that's not the whole story; the problem is not "how can we convince ourselves motion is theoretically possible?" for which the continuum view does a good job: the problem is "how can we reconcile motion with infinite divisibility?" for which switching from the infinitely divided viewpoint back to the continuum viewpoint is not an adequate resolution.

The (apparent) problem is that Zeno has written down an unending sequence of events, all of which must occur before Achilles overtakes the tortoise. There's nowhere to add "Achilles reaches the tortoise" in the list, because it keeps going without end.

Maybe a more familiar modern example might help show the problem: the age-old problem of "what does $1 - 0.\overline{9}$ equal?" ($0.\overline{9}$ means the $9$ repeats infinitely)

A naive, incorrect answer from those who haven't really grasped what infinitely repeating means is that this is $0.\overline{0}1$. These people won't see the problem that Zeno brings up, so I assume you aren't one of them.

However, once we understand what's going on, we understand that the borrow is unending; we keep getting $0$'s infinitely as we move right, and there simply isn't any place left for a $1$: we understand that the difference really is $0.\overline{0}$, or just zero.

Now, Zeno's clever argument is analogous to saying that there can be a $1$ after infinitely many $0$'s after all. And since we understand that really isn't possible, so the idea of infinite divisibility doesn't hold water. (or the idea of motion doesn't hold water, as Zeno claimed)

What we need to resolve this problem is the idea of a transfinite sequence of events. That we really can have an infinite sequence of events, and then more events after that.

Since Zeno's time, we've come up with more twists on this; if you can convince yourself that it really does make sense to look at Zeno's sequence of events and conclude that they can be completed and continue on with Achilles overtaking the tortoise, then the next puzzle is why Thompson's lamp doesn't show such reasoning to be formally absurd.

*: Zeno isn't around, so we can't ask if this really is what he had in mind.

Solution 4:

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow)...

The ancient Greeks didn't have a precise notion of speed. It wasn't until the 16th century that Galileo first measured speed by considering the distance covered and the time elapsed.

Assuming constant speeds $S_A$ and $S_T$ (m/s) for Achilles and the Tortoise respectively, we know that, in this example, Achilles would have caught up to the Tortoise in $\frac {100} {S_A-S_T}$ seconds.

With only a vague notion of speed, the ancient Greeks were perplexed by the fact, in that time interval, both racers would have passed through infinitely many points in space, the arrival at each point being an "event". In modern modern mathematics, we have no problem with infinitely many such events occurring in a finite time interval.