It turns out that in the book “The Fault in Our
Stars”, by John Green, there is a reference on Cantor’s most famous result:
there are infinities bigger than other infinities. The reference was slightly wrong, but
nonetheless alluring, and the message left a seed in the mind of one of the
protagonists (no spoilers!) How did the seed developed? Let’s hear:
I am not a mathematician, but I know this:
There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an
infinite collection of others. Of course, there is a bigger infinite set of
numbers between 0 and 2, or between 0 and a million. Some infinities are bigger
than other infinities. A writer we used to like taught us that. There are days,
many of them, when I resent the size of my unbounded set.
NOOOOOOO! This is so wrong! Completely wrong!
Abhorrently wrong. Wrong like socks and sandals. Like ketchup on pizza. Like
“Smells Like Teen Spirit” on the ukulele. Like snow in August (in the northern emisphere). Like when two people say goodbye and then they walk in the same direction. When I heard this at the cinema I unleashed all my
rightful fury (with the dismay of my fellow moviegoers. Sorry).
To be honest, John Green claims that he knew it
was wrong (attention, spoilers in the link), but he wrote it wrong on purpose, to show that teenagers can and do
reach incorrect conclusions, but find comfort in them anyway. Yet, there is no indication
in the book (or in the movie) that it is wrong, so it’s easy to imagine a girl
reading the sentence, thinking “There are more numbers between 0 and 2 than
between 0 and 1? Duh, of course. And this is what Cantor said? That Cantor guy
is useless”, then the girl becomes a researcher in the University, they ask her
to evaluate a proposal for a grant, she reads “Cantor” and think “this research
is nothing worthy”, and then I become unemployed. And nobody wants that.
So, why there is the same amount of numbers
between 0 and 1 and between 0 and 2?
First, let’s agree on the meaning of “same
amount”, with an example. Let’s go against the timeless suggestions of my
mother, and compare apples and oranges.
If Cantor has a bowl of apples and Kronecker, his
eternal nemesis, has a bowl of oranges, how can they know who has more, since
they don’t trust the counting capabilities of each other?
They pick some string, and start connecting
every apple to one orange, one by one. If in the end the bowls are empty, there
was the “same amount” of oranges, of course. (*)
And this is what we are going to do with the infinities.
If I can connect any number between 0
and 1 to a number between 0 and 2, finally without leaving out numbers, it is
clear that there is the same amount of numbers. Here we go: connect every
number between 0 and 1 with its double, like this
And so on. Did we leave out some number? Nope. Every number between 0
and 2 is connected with its half, so 0.3 is connected with 0.15, 1.33 is
connected with 0.665 and so on.
Once you see it, it is pretty clear (even for a teenager? Yes? Please
tell me so, I have such a good faith in teenagers). So that’s it: I righted the first error.
Yes, because there is another mistake! The parade of the errors seems to
have no end. Maybe. I don’t know, the problem is that I simply don’t understand
the last sentence. It’s like many pretty words with no real meaning. So, since
probably John Green won’t come here to explain this, I have to guess, and it
seems to me that the protagonist is lamenting that there are too many numbers
between 0 and whatever. But then the word “unbounded” is wrong! That set is not
unbounded, it is perfectly bounded, by whatever. But it’s infinite. Yep,
unbounded and infinite are two different things.
Another thing! Like Columbo, I want to add just a small thing before
leaving: even if the premises are wrong, in the end the protagonist is right in
finding comfort in the infinite. There is an infinity of moments even in a
short period of days, so you can project an eternity in there. So romantic
(and, most importantly, correct)!
(*) One can, in fact, refuse to use this as a method of measuring the "same amount". But then, what can we use? Some mathematicians came out with a definition such that, yes, there are more numbers between 0 and 2, but the definition is incredibly complex and unintuitive. I am open to the possibility that in the future there will be an intuitive definition that satisfy our intuition, but until then, we do with what we have. If you have better ideas, please write them in the comments!
Literature is in fact a great resource for infinity. You enjoyed apple and oranges? Here is something even more challenging, courtesy of David Foster Wallace.
And if all this infinities counfuse you, Borges has a nice way to put some order.