Have you seen the 2014 movie, “The Fault in our Stars”? It was witty and well-acted. Parts of the movie were shot in Holland, where I took this very picture. But that’s not the reason I’m talking about the movie below this picture. In the movie, one of the characters said, “Some infinities are bigger than other infinities.” Well, I haven’t done mathematics for a long time, but I know that statement is mathematically correct.
The numbers we use for counting (i.e., 1, 2, 3, 4, …) comprise an infinite set. It is not difficult to establish a one-to-one correspondence between the counting numbers and the rational numbers (i.e., counting numbers, fractions/decimals, mixed numbers, negative numbers). And yes there are as many real numbers between 0 and 1 as the set of all real numbers. If that’s not intuitive, look at the picture I took. I drew 2 horizontal lines in black color. The bottom horizontal line is longer than the top horizontal line. But if you look at the white lines, you will see that the bottom horizontal line can be mapped in a one-to-one manner to the top horizontal line, even when the bottom horizontal line is longer! Every point on the top horizontal line corresponds to a point on the bottom horizontal line. Is this surprising? Not for infinite sets. Mathematicians, if I recall, define an infinite set as that which has the same size as its proper subset.
That’s why I love taking pictures like this. I cannot have an infinite life. In fact, I already consider myself lucky for having reached this far in age. I’m entirely aware that I live in a finite world. What I can do in a given day is finite. How much money I have in the bank is definitely finite. But when I see a scenery like this, I have to say I get to experience the infinite.
So what did Peter van Houten mean when he said some infinities are bigger than other infinities? Well, as it turns out, there are more real numbers between 0 and 1 than all counting numbers. In preparation for her boyfriend’s funeral, the principal female character wrote, “I cannot tell you how thankful I am for our little infinity. I wouldn’t trade it for the world. You gave me a ‘forever’ within the numbered days, and I’m grateful.” There’s no doubt in my mind that Gus and Hazel Grace experienced infinity. What they shared was endless, boundless, celestial love, incapable of being made insignificant by the cancer that ravaged their physical bodies.
September 30, 2014