The most controversial axiom in mathematics and its paradoxes
Mathematics is based on the Zermelo-Fraenkel axioms of set theory. Among them, there is one that for a long time has been very controversial. It’s called the axiom of choice. We’ll look at what it assumes and what are its implications.
The axiom of choice
The axiom states that for every collection of non-empty sets, there exists a choice function, mapping each set in that collection to one of its elements. So let’s say we have a collection {{0,1,2}, {10}, {5,0,2}}. Then a choice function is e.g. such mapping {0,1,2} to 0, {10} to 10 and {5,0,2} to 0.
If by now you’re asking yourself why on earth is this an axiom, it means you get it. We’re going to try to make you ask why on earth is this axiom controversial. That’s gonna mean you really get it.
Indeed in many cases, we do not need to invoke the axiom of choice to get a choice function. For example with the collection consisting of all non-empty subsets of positive integers, we can map each one to its minimum element.
The problem arises with subsets of real numbers. What would a choice function be in the case of all (non-empty) subsets of real numbers? We can’t take the minimum, as the set (-∞, x] has no minimum (-∞ is not a real number). In fact, no choice function is known to exist in that case.
We’ll use Bertrand Russell’s analogy to understand why the axiom of choice is not entirely obvious.
For any collection of pairs of shoes, we can easily think of a choice function — always choose the left shoe. Now we replace shoes with socks. How do we know which one to choose?
The point is that we may have sets with elements that are indistinguishable from one another in some respects. That makes choosing exactly one of them an issue.
Banach-Tarski paradox
Many think that the Banach-Tarski theorem says we can duplicate any 3-dimensional ball, meaning we can make two balls out of one, each the same size as the initial one. But that is a fact independent of the axiom of choice, as in mathematics you can easily construct bijections between sets of different sizes. For example, x ↦ 2x is a bijection between intervals [0,1] and [0,2] with inverse x ↦ x/2.
Banach-Tarski states that we can smash our ball into finitely many pieces, then rotate and translate (move) each one in such a way that produces two balls.
Why is this harder to believe than the previous misconception? Because rotations and translations are very special kinds of bijections. They absolutely should preserve the volume. Imagine rotating and moving a certain 3-dimensional object. Where in the process does its volume change? That’s right, nowhere!
This was a real problem in measure theory, where we try to define a function that measures the volume of sets. We want it to be invariant towards things like rotations and translations. Well, tough luck.
Measure theory did eventually solve this by restricting the sets on which a measure operates. And that lead to major developments in probability theory. So, thank you axiom of choice?
But wait. Why can’t we do this in real life? If we obtain say a gram of gold, can we cut it up a bit, rotate and move the parts, then put them together into two grams of gold? Probably not, right? That’s because real numbers are anything but real - they are a perfectionist’s view of reality. There’s no get-rich-quick-scheme here.
Predicting the future
Banach-Tarski is not the only paradox stemming from the axiom of choice. There are many others. One very interesting states that we have an almost infallible strategy for predicting a real function’s value. So, given a real function f and knowing only its values on (-∞, b), our strategy correctly guesses the value f(b) on all but countably many points b.
If you don’t know what countable is, it’s the size of positive integers. That might sound big (is in fact infinite), but it’s negligible in comparison to the size of all real numbers. What do I mean by that? The standard measure of the real interval [0,1] for example is 1, while the same measure of all integers is 0. Why does that matter? In the standard model that implies the probability of us picking an element from a given countable set is 0.
Translating this directly to the paradox, it says that we can predict any function’s value only by knowing all the previous values, with the probability of an incorrect prediction being 0.
That’s pretty insane. I mean, a function doesn’t need to be dependent on its previous values AT ALL! It’s not a continuous function, it’s ANY function.
Why keep the axiom of choice?
Why keep any of the axioms for mathematics we have? To make mathematics stronger. With more axioms, we can prove more things. Some of those being close to contradictory might just be a price to pay.
The axiom of choice makes proving things not only possible but also easier. The whole theory of functional analysis is built on the Hahn-Banach theorem, the proof of which requires an equivalent to the axiom of choice — Zorn’s lemma. Abstract algebra often relies on the fact that every vector space has a basis, or that a maximal ideal always exists. Those are again consequences of the axiom of choice. There are countless examples of just how integral it is to some theories in mathematics.
Also, it turns out we just can’t escape having some paradoxes. Without the axiom of choice, we can have sets with sizes that cannot be compared. Two sets can be neither the same size nor different size. This goes away when the axiom of choice is included.
Finally, there is a result guaranteeing us safety. It is proven that the axiom of choice doesn’t affect the consistency of math, meaning if math is contradictory, it’s not the axiom’s fault.
Conclusion
We’ve seen some pretty insane claims that are called paradoxes. In math, something is called a paradox if it’s very counter-intuitive. It doesn’t mean it’s contradictory to mathematics in any way. Paradoxes make us reconsider our intuition and see just how crappy it can be. So the lesson here is, if you encounter something paradoxical, think it over. It might just require a bit more understanding.
