Infinity

So imagine you're Usain Bolt. You're like six and a half feet tall. You have a couple Olympic medals—no biggie. You show up to your final Olympic race, and your only opponent is a tortoise. For some reason, he has gold medals around his neck. He can talk and he challenges you to a hundred meter race. You're confused, but you accept his challenge.

His only condition is that he gets a 50 meter head start. I mean, seems fair, right? He is only a tortoise. So you'd line up at the start of the race, and the gun goes off. The tortoise begins his 50 meter head start. It's gonna take a while, so feel free to get a coffee, go home, take a nap—just do whatever you want.

After a while, he finally gets to the 50 meter mark, and you sprint to catch up with him. But in the time that it took you to get to the 50 meter mark, where the tortoise was, he's moved forward another 10 meters. No problem; you just have to catch up to the point to where he is now. But wait! Once you get to that point, the tortoise has moved forward another five meters.

So you have to catch up again. This process continues to repeat and repeat an infinite amount of times. We can continue cutting the distance between you and the tortoise in half as many times as we want. It'll take a finite amount of time to complete, but the distances can continue to be cut in half forever.

Let's say it takes you five seconds to run the initial 50 meters, then another one second to run the extra 10 meters, then another half a second to run the next five. All of these times are finite, but there's an infinite amount of distances you have to travel. So by this logic, it should take an infinite amount of time to catch up to the tortoise, right? You can never catch up to the tortoise.

In the end, he beats you in the race. It makes no sense, but this logic says you literally cannot pass him without doing more than an infinite amount of tasks. Not only can you not beat the tortoise, but you can't move anywhere without doing an infinite number of tasks. Thus, movement from your bed to your refrigerator should take an infinite amount of time.

So where's our math wrong, or is motion impossible? [Music] Clearly, this argument is insane. I mean, race any tortoise in the world, and you'll beat it in a race every single time. But where's the flaw in the logic?

This is known as one of Zeno's paradoxes. There are multiple, but they all are just the same thing: motion, as we know it, is an illusion. The thought is, in order to finish the race, or really any movement in general, you'd first have to get halfway between your starting point and the finish line, which in the race's case is 50 meters. From there, you'd have to get to the halfway point between the 50 meter point and the finish line, the 75 meter point.

This continues being cut in half an infinite amount of times. This idea can also be reversed, and this is where things get interesting. If we reverse the sequence, what number comes first? We can't start at 50 because we can divide that by two. We can't start at 25 because we can divide that in half as well. In fact, we can divide any finite number in half an infinite amount of times.

This means that there is no first distance to run, therefore making motion impossible, or maybe not impossible, but just an illusion. The idea was that because there's an infinite amount of distances being added, they must take an infinite amount of time to complete, right? Not really. Zeno was wrong to assume that there's an infinite amount of distance to traverse.

Calculus kind of solves the problem. With the example of the 100-meter race, cutting their distance in half each time just creates what is known as a convergent series. When you add all of the infinite amount of terms in this series together, you don't get infinity; you get 100—a total distance of the race. All of the terms, when added together, converge to 100. This solves the problem of infinite distance.

But what about the race versus the tortoise? We obviously know that you'd pass the tortoise during the race. But how? If you run at 10 meters per second while the tortoise only moves at 2 meters per second, well after his 50 me...