Graphic Artist: Margaret Cartee

I stubbornly resisted watching John Green’s The Fault in Our Stars until a midnight flight back from an 8^th grade field trip. Sitting in the middle seat between two pre-teen boys, I dutifully rolled my eyes at the mushy dates, poignant declarations, and tragi-romantic ending. Even if you haven’t seen the movie, most people remember Hazel Grace’s pre-eulogy delivered to her terminally ill boyfriend Augustus. Pondering their deep romance and its untimely demise, she claims that there are more numbers between zero and two than zero and one. She famously laments some infinities are bigger than other infinities, a beautiful yet erroneous idea. Though some jump to discredit her mathematically flawed statement, I hope to remedy her logic through sets and their sizes, called cardinalities. Georg Cantor’s foundationalist set theory ultimately dictates principles that prove how there are as many numbers between zero and one as there are zero and two as well as zero and a million. Building off his principles lets us count to an infinite number or even claim there are just as many even integers as there are all integers.

To analyze Hazel Grace’s complex metaphor between mathematics and mortality, we’ll turn to the world’s arguably most renowned mathematician, Count von Count of Sesame Street. The Count uses natural numbers starting from one to quantify objects for young viewers (Figure 1). Let’s say the Count needs to know how many bats are in a group, called a set. He bijects the natural numbers zero through ten onto his friends and concludes he has ten bats. This bijection refers to the one-to-one correspondence with or bijection from each of the natural numbers up to ten onto each of his bats. Ten therefore represents the cardinality or size of the set [1]. The Count now visits Dracula and his twenty bats. Though Dracula has twice as many bats, von Count makes another bijection of the natural numbers zero through twenty onto Dracula’s bats. The cardinality of von Count’s bat set is ten while Dracula’s cardinality is twenty. For fun and unintentionally for later, the Count decides to count every other one of Dracula’s bats to confirm that Dracula has more bats. Von Count’s finite sets behave intuitively even to his youngest, most avid fans. However, what if he faces an extremely, inordinate number of bats beyond a Sesame Street episode?

Von Count travels to Batman’s Batcave, which houses as many bats as there are integers. As they fly out of the Batcave single file, von Count bijects the set of natural numbers onto the bats. Exhausted from all that counting, von Count decides to count every other bat as they return and doubles this value to sum the bats. If he could double his bats to compare with Dracula’s twenty bats, why wouldn’t it work with Batman’s integer set of bats? To a mathematician, von Count ponders whether bijecting onto the set of all integers then halving is the equivalent of bijecting onto the set of all even integers. To us, he ponders whether there are exactly half as many even integers as there are all integers. However, the Count finds that there are just as many even bats as there are total bats. Where did all these ‘new’ bats come from? The difference between Dracula and Batman’s situations distill down to how mathematicians classify types of sets. Dracula possesses a countably finite set with a cardinality of twenty bats. Batman, however, has a countably infinite set of bats with a cardinality of all the integer numbers. This term seems paradoxical: how can we count to an infinite number? And how is a sub-set of even integers the same size as the full set of all integers?

The befuddled batmen seek out mathematician Georg Cantor to explain set theory. Cantor postulates that von Count could theoretically count Batman’s bats with natural numbers, but it must have taken him forever. As von Count successfully bijected a natural number onto all of Batman’s bats counting both every single and every other one, Cantor claims that Batman has aleph-null (ℵ₀) ‘number’ of bats. Yet, aleph numbers are not true numbers like von Count’s twenty bats, integers like 0, 1, 2, 3…etc. π , e or any imaginary number and certainly don’t behave like them (Figure 1). Aleph numbers describe the cardinality of sets with sizes that exist beyond any large, finite countable number. Aleph-null represents the smallest, first aleph number where you can biject a natural number onto each element of your set. Cantor proved that the set of all integers has the same aleph-null cardinality as the sub-set of all even integers. In other words, there are indeed just as many even integers as there are all integers. His work establishes:

demonstrating aleph numbers and their algebra do not behave like classical finite numbers. By the second line, it doesn’t matter if von Count counts every single, other, third, fourth, or tenth bat: Batman always ends up with aleph-null bats simply because he’s Batman.

Where does this leave Hazel Grace? Unlike the sole infinity, ∞, some aleph numbers are indeed larger than others. While Dracula and Batman have finite and countably infinite sets, Hazel Grace’s set extends beyond countably infinite. The batmen biject natural numbers onto concrete objects like bats you, me, or a toddler can physically count. However, Hazel Grace’s numbers between integers zero, one, two, or a million delineate number lines with decimal values intermediate to these whole numbers (Figure 1). There’s a stark difference between the batmen’s natural numbers and Hazel Grace’s real numbers. As opposed to aleph-null, the next smallest cardinality ℵ₁, aleph-one, describes this cardinality of a set of all real numbers between any two for number line sets [3]. Mathematicians later proved 2 ^ℵ₀ or that 2 raised to the power of aleph-null, the number of natural numbers, gives aleph-one. Put simply, there are more real than natural numbers. The Count can biject natural numbers onto the integers to derive aleph-null, but he would run out of integers to biject onto the real numbers. Assuming Hazel Grace’s first set between zero and one has aleph-one cardinality, Cantor’s equations state that there are just as many real numbers between zero, one, two, or a million. Sadly, scientists reserve infinity to describe functions’ end behavior, boundless limits, or projective geometry, not love. That is unless any of those topics scream love to you.

If von Count, Batman, and Cantor wrote The Fault in Our Stars, Hazel Grace would opine that some cardinalities are bigger than others, even though her sets between zero, one, two, and a million all have size aleph-one (ℵ₁), the cardinality of real numbers. Because ℵ₀ < ℵ₁ < ℵ₂…ℵ_n, she’s now correct that some relationships and cardinalities measured by aleph numbers are larger than others. Aleph numbers may not wax as romantic, but they give her statement a more optimistic perspective. Extending this metaphor between relationships and cardinalities implies it’s always possible to find another just as meaningful. There are many aleph numbers compared to the traditional concept of a singular infinity. Some relationships are countably short, others countably infinite or truly uncountable. Yet, each is distinctively unique for the numbers and memories they contain. Even if both time and aleph-numbers face some distant unseen end, such limits should encourage us to celebrate finite moments and the people within them in the great sets of life.

Acknowledgements:

I would like to thank my professor Dr. Jeffrey Yelton for reviewing, editing, and guiding my work.

References:

1. Lamb, E. (2014, July 10). The Fault in Our Stars’ Faulty Math. Scientific American Blog Network. Retrieved March 6, 2022, from https://blogs.scientificamerican.com/roots-of-unity/the-fault-in-our-stars-faulty-math/
2. Wolfram MathWorld. (2022). Aleph-0. Retrieved March 6, 2022, from https://mathworld.wolfram.com/Aleph-0.html
3. Lian, T. (2011, August 23). Fundamentals of Zermelo-Fraenkel Set Theory. UChicago Math. Retrieved March 6, 2022, from https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf