THE AXIOM OF CHOICE
by Andy Boyd
Today, we make a choice. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
Imagine the following scenario. My dresser has nine drawers. I personally guarantee that every drawer has one or more shirts in it, and I ask you to choose one shirt from each drawer for me. Can you do it?
A common dresser. Photo Credit: Andy Boyd
If it seems obvious that yes, of course you can, you’re not alone. But starting in the early twentieth century a similar question caused waves of controversy in the mathematical community. At issue were the very underpinnings of math itself. Mathematicians had grown uneasy that the foundation upon which the vast mathematical edifice had been built wasn’t as solid as it needed to be. So they went back to basics, which meant sets. The set of cups on a table. The set of shirts in my top, left dresser drawer. The goal was to state the most basic assumptions, or axioms, about what can intuitively be done with sets, then show how all of math can be built on those axioms.
In 1908 a young German mathematician named Ernst Zermelo proposed a collection of seven axioms. One, known as the axiom of choice, was the same as our intuitive assumption about the dresser drawer problem. The axiom states that given a collection of distinct, non-empty sets you can always choose an item from each one. Zermelo was saying it’s so obvious we can get a shirt from every drawer that he was going to include it in the very foundations of math.
Ernst Zermelo. Photo Credit: Wikimedia
Not so fast said other mathematicians. It’s important to provide a description of how each choice was made. For example, when I asked you to get a shirt from each of my dresser drawers I guaranteed each drawer had a shirt, but I didn’t tell you how to find one. Maybe the drawers are huge and filled with nooks and crannies making the task difficult or impossible. This becomes a special issue for the dressers mathematicians work with, where the drawers may be infinitely large and the shirts infinitely small. Since mathematicians were seeking a simple yet strong foundation for math that held no surprises, the status of the axiom of choice was a big concern.
Some of the finest minds in history weighed in on the topic. Was the axiom necessary? Did it lead to logical contradictions? And was it obvious? It was as close to a purely philosophical question as mathematicians would ever come. And the eventual answer was a bit surprising.
Today the most commonly accepted axioms underlying all of mathematics are a modified version of Zermelo’s original axioms. But these axioms come in two distinct flavors: one with the axiom of choice and one without. Take your pick. It’s a matter of personal choice. Both lead to perfectly good foundations for math, though the additional power of the axiom of choice allows us to go farther and do things more easily. But if you’re uncomfortable with the thought of finding shirts in dresser drawers without explicit instructions, not to worry. We’ll just settle on the fact the shirts are there and leave it at that.
An open drawer. Photo Credit: Andy Boyd
I'm Andy Boyd at the University of Houston, where we’re interested in the way inventive minds work.
The most commonly accepted axioms are the Zermelo-Fraenkel axioms with and without the axiom of choice, abbreviated ZF and ZFC, respectively.
The discussion of the “best” set of axioms continues to this day as mathematicians attempt to balance two competing goals. On one hand, the axioms should be strong enough to allow all “important” mathematical results to be derived. On the other, they should not be so strong that they allow “strange” results to be derived. Of course, “important” and “strange” aren’t well-defined concepts, varying with the temperament of the individual who’s doing the judging. The axiom of choice gives rise to some results many mathematicians consider strange, or at least highly counterintuitive. Axioms other than ZF and ZFC have also been proposed, all with their various strengths and weaknesses.
J. Bell. “The Axiom of Choice.” The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta, ed. See https://plato.stanford.edu/archives/sum2015/entries/axiom-choice/. Accessed January 3, 2017.
This episode was first aired on January 5, 2017