In the not entirely
mathematical romantic drama “The Fault in our Stars” one of the main
characters- Hazel Grace- provides us with a nut of mathematical wisdom: “There are infinite numbers between 0 and 1. There's 0.1
and 0.12 and 0.112 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.” Now lovely as this is, is it
mathematically correct? In this piece I will discuss that in what I like to
call the fault in The Fault in Our Stars.
So to
decide if one infinity is bigger than another you need to ask easy questions to
answer harder ones. How do you know one set of things is bigger than another?
Now this may seem trivial however it is an important question for we cannot
simply count both sets and say which is bigger in this case as they are both
infinite. So instead we do this a different way. Imagine there is a shepherd
who has never had the joy of sitting in a single math lesson. As a result he
has no knowledge of numbers, so how can he count his sheep when they go into
their pen? Well he may struggle and just guess on what it looks like if he has
all his sheep but he is very likely to get it wrong. Instead he should place a
stone by the gate for every sheep that passes him. Then by matching each sheep
to a stone he can check he has them all by removing a stone as each sheep
passes him. By pairing up stones and sheep he has made two sets of equal size!
We can do
a similar thing with sets of numbers to see if two sets are the same size. The
step of pairing stones and sheep in this case is different. For sets you must
come up with a function that matches every member of one set to only one member
of the other. This function must also include all members of both sets. For
example if we wanted to see if there were the same number of even numbers as
odd numbers. By applying the function of adding 1 to each odd number we get
every even number. Therefore we can say there are as many even numbers as odd
numbers. This is exciting! We now have a way of effectively seeing if two sets
are the same size. Now say if we were to explore if there are as may positive even
numbers as natural numbers. This seems ridiculous but you can see that if you
double every number you get an even number. Therefore we come to the remarkable
conclusion that they are the same size! Just as the shepherd in this case we
would have been likely to get this wrong by just looking at it.
So the big
question. Is Hazel Grace right? Well there are an infinite number between 0-1
and an infinite number between 0-2. But if you double all the numbers between
0-1 you end up with every single number between 0-2! This leads to the surprising
and counter intuitive conclusion that
there are as many numbers between 0-1 as there are between 0-2.
But are some infinities
bigger than others? Well imagine we make an infinite list of all the natural
numbers so 1,2,3… This is known as countable infinity but listable may be a
better name .The point is that a system can be made that would list all of the
numbers without missing any out. However say you tried to make a list of the
numbers between 0-1. If you think you have a list it will always be possible to
create a number that isn’t in your list. You do this by changing the first
digit of the first number and the second digit of the second number and so on
for an infinite number of digits and numbers. You could simply add 1 to each
digit and if it is 9 change the digit to 0. Now you will have constructed a
number which is different to every number and therefore not on the list. You
could vary the digits an infinite number of ways therefore your infinite list
is infinitely small when compared to the infinite infinity of real numbers!
Phew, what a mouthful… But forgetting all the confusing sentences, this shows
some infinities are bigger than others.
So to conclude there is
a fault in the Fault in our Stars and Hazel Grace may need to read up on her set
theory!
Jake
