Science
fiction and ”serious and intellectual” novels sometimes contain references to
Set Theory, but starting the blog with them would be cheap. Of course there are
references, these writers are geeks, what do you expect? I want to begin with a
blast! Somewhere unexpected! So, what about… a young adult romance novel?

John Green
is a popular writer of young adults novels, mostly heartbreaking teen romances,
and he’s pretty good at it. If you like the genre, it doesn’t get any better
than that. Despite the constraints of the genre, he always tries to put
something new in its stories. My
attention goes to “The Fault in Our
Stars”, a novel published in 2012, and then adapted in film in 2014. And yes,
he went there. This is an excerpt:

“So Zeno is most famous for his tortoise
paradox. Let us imagine that you are in a race with a tortoise. The tortoise
has a ten-yard head start. In the time it takes you to run that ten yards, the tortoise
has maybe moved one yard. And then in the time it takes you to make up that
distance, the tortoise goes a bit farther, and so on forever. You are faster
than the tortoise but you can never catch him; you can only decrease his lead.

“Of course, you just run past the tortoise without contemplating the
mechanics involved, but the question of how you are able to do this turns out
to be incredibly complicated, and no one really solved it until Cantor showed
us that some infinities are bigger than other infinities.”

So, is this
correct or not?

**It goes very, very close**, I am tempted to say that is correct, but

**no, it isn’t**.

Let’s start
from the beginning: Zeno’s paradox of Achilles and the Tortoise. The quote does
a good job to illustrate it, down to the big problem.

**Why “you can never catch him”?**Because it would take an infinite amount of intervals of time? Or of space? Because the sequence never ends?
Anyway,

**all of this was solved in the 19**, during the complete overhaul of Analysis that has been made by, for example, Weierstrass and Cauchy: it simply is possible to have a sum of infinite terms that gives a finite result. So Achilles will catch the elusive tortoise in finite time (or space).^{th}century
Problem
solved? Nuh-uh. You wish. Yes,

It was Aristotles fault, of course, like it always is: he practically said that the only way to think about infinity is to think of finite sets larger and larger, but we cannot deal with an infinite set.

**the solution made perfect sense and the calculations worked, but it was not sound, because everybody was scared by infinite sums**. “Infinity is just for God! How can we puny humans deal with this!”
(Imagine Kronecker with a shrill voice saying this)

It was Aristotles fault, of course, like it always is: he practically said that the only way to think about infinity is to think of finite sets larger and larger, but we cannot deal with an infinite set.

And here
comes our hero Cantor to save us, daiquiri in hand! With his study,

**he proved that it was mathematically sound to deal with infinite sets**, therefore giving the theoretical support necessary to Analysis to solve Zeno’s paradox. I am not saying this, Russell said it. Also, at the same time with his brilliant work he canceled centuries of superstition, advanced the knowledge of the entire humanity and validated all the smart kids that to win at the game of the biggest number shouted "Infinity plus one!"
So, Zeno
-> Cantor, right? Where is the error? Well, John Green quotes Cantor’s
Theorem: yes, even if it goes against our
intuition, there are infinities bigger than other infinities. This is the
starting point of modern Set Theory, the pedestal upon which the magnificent
crystal castle is built. The proof of this is practically everywhere, it is
called Cantor’s Diagonalization. But!

**It is not the solution to Zeno’s paradox. His treatment of infinity (that lead him to his Theorem) is.**
Ok, I admit
it, that was very nitpicky. I forgive John Green, because he couldn’t write the
history of 19

^{th}century mathematics in one line, right? But I won’t forgive what he did next, because, believe it or not, that was not the only occurrence of infinity in the book. Something much worse is coming…
Stay tuned for the next episode!

*Update: in another post I explain in more details why Cantor's approach solves the tortoise paradox. If you are unsatisfied by the explanation above, go here. And there is also Buzz Lightyear! But I stress: Cantor's Theorem has nothing to do with it.**You're wondering why John Green made this error? The behind-the-scenes answer is here!*
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