So I recently read The Fault in Our Stars by John Green. Seeing all the hype, I wanted to see what was so great about this new young adult romance comedy novel about two pretentious teenagers with cancer (but wait, it’s only a “side effect of dying”) that fall in love. Although they claim to be different from all those other kids with cancer… really, they’re not. The book is distasteful, its depiction of cancer is horrible, and most despicable of all, it has (gasp) a mathematical fallacy.
I’m kidding. It was a good light read.
Except for that last part, I’m not kidding about that. One of the biggest things that bothered me enough to make me stop chuckling at these terrible (in a good way) jokes for a while was the use of the line, “Some infinities are bigger than other infinities.” First off, the line was introduced completely out of the blue because Green wanted to make the characters look smarter and more pretentious (this also happened with Maslow’s hierarchy of needs). But aside from that, Green seriously didn’t do his research on Cantor, which isn’t too impressive considering he used it extensively to make a point in one of the passages.
To sum up that part (spoiler alert!), Hazel is delivering a eulogy for Augustus, in which she claims that there an infinite set of numbers between and , but there is a bigger infinite set of numbers between and . She feels that her situation is like that – the “unbounded set” (yet another mathematical fallacy – it is a bounded set) of time she has with him, despite being smaller than normal, provides her with a “forever within the numbered days.”
That’s great, Hazel. And actually, that’s really cute, even touching. Except it’s wrong.
Cantor’s set theory, especially that involving cardinality (i.e. size) of sets, shows that, in fact, the infinite set of numbers between and and the infinite set of numbers between and are the same “size”. To expand on that, any two infinite sets of numbers have the same cardinality if there exists a bijection between them. What does this mean? To demonstrate, we use an example with two finite sets:
Say Alphonse, Bryan, and Charles want to ask Alicia, Betty, and Claire to prom. We know that there will be enough girls for the guys to each get one partner because there are three boys and three girls – but how can we know that without knowing how many elements there are in each set? (Remember, we’re dealing with infinite sets!) The simplest way would be to pair each guy with a girl, and if there are no guys or girls left over, then the two sets have the same number of people!
(Interestingly, I read somewhere that this method was used by farmers in the past when counting wasn’t invented: to ensure that they weren’t missing any animals when they brought them in during the night, they would tie each animal to a separate stake; if there was a stake with a missing animal, they would then know that they had fewer animals than they were supposed to!)
This post is getting a bit long – I’ll continue this blog post here…