Uncountable Infinities

by Rev. Jonathan Rogers

Delivered at Northwest UUC

March 11, 2018

I learned about the concept of uncountable infinities when I was 16 years old, taking Dr. Ben Mathis’ Intro to Abstract Math course at Colby College, which was near my house growing up. I was a huge dork as a high school student, and one of the ways I would keep my mind occupied in class when I got bored was to count upward by perfect cubes. 1, 8, 27, 64, 125, 216, 343, 512, 729 (which is still my favorite number), 1000, 1331, etc. I was also, during my senior year, captain of my school’s math team. When those are your credentials as a teenager, pretty much the only place you can go and feel like one of the cool kids is your local UU congregation. The fact that the Our Whole Lives comprehensive sexuality education program we use in our congregations is abstinence-based felt very redundant to me.

The year before I took abstract math, I had taken calculus at Colby also, but I was more interested in taking language classes if given the chance. At the end of the year, the calculus professor told me that languages make the world go ‘round, but math is where the money is. I went on to study Chinese in undergrad. My professor’s point was driven home a few years later when I calculated that my salary as Youth Programs Coordinator was 1/18th of what my friend was making as a computer engineer on the search team at Google. I chose to look at the situation as both of us having the freedom to do what we loved.

One afternoon I was in Abstract Math class and Dr. Mathis claimed to be able to demonstrate that you could have two infinite sets of numbers, one of which was provably larger than the other. I was very skeptical, because after all, infinity is infinity, right? Wouldn’t there always be more elements in an infinitely large set, such that it would be impossible to prove another set had even more elements? The person who laid out a very elegant proof of the idea of different-size infinities was Georg Cantor, a German mathematician who invented set theory, and wrote the proof in question in 1891. It is commonly known as Cantor’s Diagonal Argument.

To understand Cantor’s proof, we first have to understand the concept of bijection. Bijection means that any element in one set can be paired with a corresponding element in another set. For example, it is possible to biject the set of natural numbers (1, 2, 3, 4, etc.) onto the set of even numbers (2, 4, 6, 8). You can use the function f(x)=2x for any positive integer, any element in the set of natural numbers, to biject that element onto a single corresponding element in the set of even numbers. So, even though it seems intuitively like there should be twice as many natural numbers as there are even numbers, it’s actually possible to prove that the sets containing those respective numbers are the same size, or, in mathematical speak, that they have the same cardinality. It is possible to biject every number from the set of natural numbers onto an element in the set of even numbers.

So, is it possible to biject the elements of an infinite set of numbers onto the elements of any other infinite set? Cantor showed that it is not, and not only that, but he did it in binary, using only ones and zeroes. In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). So the first sequence might be (1, 1, 1, 1…), the second sequence could be (0, 0, 0, 0…), the third set could be (1, 0, 1, 0…), and so on. And for the set T, which contains all these sequences, to be “countable,” you need to be able to say that each of these sequences can be mapped onto an element of the infinite set of natural numbers, i.e., that you can label all the sequences s1 s2 s3  . . . and any of the sequences will have a label.

So Cantor said, OK, now here is a set s, just plain old s, and its definition is going to be that each of its elements is the opposite of the n-th element in set sn. So if the first number in the first sequence is 1, the first element in s is going to be 0, if the second element in the second sequence is 0, the second element in s is going to be 1, if the third element in the third sequence is going to be 1, the third element in s is going to be 0, and so on. What one is creating with this sequence, s, is a sequence that by definition is different from any of the other sequences in the set of infinite sequences of binary digits. It would therefore NOT be possible to biject the set of natural numbers onto this set of sequences, because there was this sequence s that definitionally would be one element different from any sequence that COULD be bijected onto an element in the set of natural numbers. For a set to be countable, you have to be able to biject each of its elements onto one of the natural numbers, 1, 2, 3, all the way up to infinity. But Cantor had proved that a set could exist and have an element that by definition could not biject onto one of the natural numbers. He had proved that a set could be larger than a countably infinite set.

Furthermore, he went on to demonstrate that some sets can have infinite elements in them, each of which in turn can be demonstrated to have infinite elements. Such sets are said to have an uncountable infinity of elements, and all infinite sets are somewhere on a continuum of cardinalities between countable and uncountable infinities. AND there can be no “set of sets” that just includes all of the above, because you can also prove that any set including all possible sub-sets is bigger than a set itself, so it is always possible to create a set with more elements. So, when I am in a meeting and decisions are being made, I try to remind myself that it is important to compromise, because even if I literally knew infinity things, there would still be things I do not know.

Cantor’s diagonal argument is a very slick, clever, elegant proof, and I remember leaving the classroom that day struggling to comprehend it, to get my head around it, and I wanted to tell my dad when I got in the car with him. He would give me a ride back to my high school after class, and when I told him about this concept he looked at me and he said “aren’t they all part of the one greater infinity?” My first reaction to that was that I had some of the most intense nerd rage I have felt in my entire life! But I took some breaths and calmed down a little bit, and I could definitely see his point, that at some level there is a universe that we and everything that exists are a part of, infinities and all. What I most appreciated about his question and our dialogue was that we were able to talk about some of the ultimate matters of our world and our universe, which are not always easy to get at in a Unitarian Universalist framework where easy answers rarely exist.

In some of our congregations, that is a culture that really exists, where people feel like they can explore questions of ultimate importance from different perspectives and have a combination of mutual education, helpful consensus and friendly disagreement. And that is a great thing to be able to aim for. I remember talking with Rev. Kim Palmer, a minister affiliated with UU Metro Atlanta North over in Roswell, about this idea of uncountable infinities. Her take was totally different from mine, but I loved it, she said that if we are in a universe that is cooling as it expands, and that is possibly becoming less advantageous to life as we know it in the process, she is comforted by the notion that even that expanded universe might not be all that there is. I appreciated her perspective and her illuminating another path to thinking about this. Rev Palmer is also a Star Trek fan, and if I weren’t already so far over my nerdiness quotient for this sermon, I would tell you about our disagreements over the implications of the kobayashi maru and the prime directive.

The fifth source of Unitarian Universalism is “Humanist teachings which counsel us to heed the guidance of reason and the results of science, and warn us against idolatries of the mind and spirit.” My hope for our congregations is that we will learn both humility and an empirical approach from science. Ideas like uncountable infinities can help us to keep open minds and be humbling reminders of the inherent limitations each of our perspectives have, and that they would have even if we were somehow capable of knowing infinity things.

At the same time, we can by no means take for granted the empiricism that science is based on, an empiricism that calls us to advocate for the vast capacity of stem cell research to unlock answers, even when it is not politically favored. It is an empiricism that calls us to recognize the fallacy of genetic racial superiority by demonstrating that humans have much greater genetic differences within racial categories than between them. Empirical data and science should by no means be the sum total of how we know the world around us, and indeed part of the beauty of science is that it is always raising more questions than it answers. But we should at least be basing our views of the world on the scientific answers that we DO have. We should be aiming to get UP to data, not ignoring it because it can’t answer all of our questions. Science and data ought to be both informing us and our decisions, and also inspiring the vast amounts of awe and wonder that its concepts and conclusions evoke. I try and encourage young folks now to get a healthy dose of both the informing and the inspiring so that they don’t end up making 18 times less than their classmates.

Formulas and equations are matters that can be written in textbooks and put on the shelf until the next time we need them. E=mc^2 is something we can look up any time. But things like our principles, and our sense of awe and wonder must be practiced regularly, or we will lose them. That’s why we make a ritual and a practice of coming together each week to remind ourselves and each other of what it means to have and to embody a sense of awe and wonder.

Like Rev. Rosemary Bray McNatt, the President of the Starr King School for Ministry, let us come because both Spirit and reason are dear to our hearts. Let us be here because we can make mistakes while still being held accountable for doing our best. Let us be here because every time we think our set of sets contains all the sets, something happens to prove to us that that is impossible for us as humans, and to re-open our eyes in gratitude to the transcending mystery and wonder of our world and of our universe. May we each be filled with this gratitude and wonder all the days of our lives!

Peace, salaam, shalom, and may it be so.