Explaining Infinite Sets and The Fault in Our Stars

Question

In watching The Fault in Our Stars I could not help but cringe at a line that flew in the face of mathematics and subsequently ruined the movie for me:

"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." - John Green

While walking out of the theater I tried to explain to my friends why there were, in fact, exactly the same amount of numbers between 0 and 1 as 0 and 2, but Cantor and bijective functions are not great learning tools to English majors.

Does anybody have an eloquent or elegant way to enumerate this phenomenon using an example accessible to those not familiar with advanced mathematics?

Answer 1

Assume Alice has a basket with balls in it, one for each real number between $0$ and $1$, which is written on the ball. OK, it is hard to imagine so many balls - or even how one would manage to write down an arbitrary real number on such a ball, but that is not the point here.

Bob also has such a basket, also with one ball for each real number between $0$ and $1$. If there is any concept to make sense of this at all, we can only say that Alice and Bob have the same number of balls in the basket.

What if Bob took out his marker pen and painted a green dot on each of his balls? Of course, the two buddies still have the same number of balls.

What if Bob instead would prepend the symbols "$2\times$" before the number written on the ball? Of course, the two buddies still have the same number of balls. The difference to the previous example is minor - green dot or a digit and a times symbol, that does not make a difference.

What if Bob now for each of his balls replaces the number ($x$, say) written on it with the number $2x$? Of ocurse they still have the same number of balls. The difference to the provious example is minor - whether the ball has "$2\times0.314159$" written on it or "$0.628318$" doesn't matter.

Now on closer look, Bob notices that in his basket he has one ball for each real number between $0$ and $2$.

We conclude that there are just as many numbers between $0$ and $1$ as there are between $0$ and $2$.

Answer 2

Look, don't worry about it. The author is absolutely correct if by "bigger" he means bigger Lebesgue measure rather than bigger cardinality. Cardinality is just one way to abstract our intuitions about size and it isn't obviously the best one to use in all situations (especially in this kind of situation where it returns highly counterintuitive results).

Answer 3

Draw two parallel line segments, a small one and a big one, then construct the triangle formed by uniting their end-points. Notice now how to each line determined by the newly-obtained vertex and a point on the small segment there corresponds exactly one point on the bigger segment, and vice-versa, thus proving that the two have the exact same number of points, regardless of their different lengths.

Answer 4

If you drive $60$ kilometers per hour you will traverse one kilometer in one minute. If you drive $120$ kilometers per hour you will traverse two kilometers in one minute. In this way each distance in between $0$ and $1$ kilometer correspond to a time in between $0$ and $1$ minute. But, the same is true for each distance between $0$ and $2$ kilometers.

Answer 5

For any number between 0 and 2, you can divide it by 2, and get a number between 0 and 1. If you divide two different numbers by 2, you will always get two different answers. Thus, there exists a one-to-one correspondence between the set of numbers between 0 and 1, and the set of numbers between 0 and 2. This means they have the same cardinality.

This argument is basically a less verbose version of the "balls in basket" one, and a less geometrical version of the one with the parallel lines, but I've had success explaining it to English majors in the past.

Answer 6

Think of a group of girls, and a group of boys. Which group is bigger? We can match them up in pairs to see. If we have one boy for every girl and vice versa, then the number of girls is the same as boys.

So it is with the numbers. Take any number in between 0 and 1, and multiply it by 2. That gives a number between 0 and 2, so we have paired up numbers between 0 and 1 with numbers between 0 and 2.

Answer 7

I actually think being trained mathematically makes us instinctively use cardinality to decide if a set is bigger than another one. But in a more unsophisticated sense it is perfectly reasonable to say that if an infinite set $A$ is a proper subset of the infinite set $B$ then $B$ is larger than $A$. For example, suppose you are using induction to show some identity holds for every natural number $n$. If you only show it for even $n$, then you are perfectly reasonable to believe that you have only done half the job. The fact that the even natural numbers have the same cardinality as all natural numbers would not necessarily make you think otherwise... even though a few minutes later you'd angrily complain that the movie has got it wrong.

So maybe we are all being wrong by automatically using cardinality as the indicator of the size of a set as the default in everyday life (such as teen romance films.)

Answer 8

I think illustrations work wonders when you are trying to explain and talk about abstract concepts like cardinality.

So for example I would suggest the following one to show that there are as many real numbers as there are between two.