Irregular Webcomic! #4035

I've talked about the Banach-Tarski theorem at length before (even giving an outline of the proof).

Briefly: The Banach-Tarski theorem is a mathematical theorem that proves that you can decompose a sphere into a finite number of pieces, and reassemble those pieces to produce two spheres of the same volume. I won't dwell on it further here - follow the link if you want to know the details. But yes, that's what it sounds like: a proven mathematical theorem of geometry that says you can cut a solid object apart and reassemble it into two exact copies. Clearly something Jesus would have found useful to feed the multitudes with a few loaves and fishes.

Okay, back in that outline proof of the Banach-Tarski theorem, there was a bit that said:

Make a new set of points, call it M. The set M is made up of exactly one point from each of the slices we've just defined. It doesn't matter which point we take from each slice, as long as we take one point, and only one point, from each slice. (We can do this by using the axiom of choice, assuming you believe it - but that's a story for another annotation.)

This is that annotation.

An axiom is a premise or postulate of mathematics or logic. It is ideally something that is so elementary and so obviously true that there is no need to try and prove it. The entire edifice of mathematics is then built on its axioms, using nothing but the axioms and formal logical proof to derive other true results. For example, using the accepted axioms of mathematics and geometry, you can prove the Pythagorean theorem.

Sometimes an axiom is something that is so obviously true, but which cannot be broken down and proven using other axioms, that mathematicians decide that it must be counted as an axiom and assumed to be true. Euclid did this when he tried to formalise the study of geometry. He tried to prove the obvious fact that parallel lines never intersect, but found that he couldn't do it using his proposed list of axioms about geometry. So he decided that the fact that parallel lines never intersect must be an axiom, added it to his list of geometry axioms, and went on his merry way. Which was indeed fine for the familiar sort of geometry that we learn at school. But in a wider sense, Euclid made a huge mistake. If you assume the opposite of Euclid's axiom - that parallel lines do meet, then you can develop an entire self-consistent theory of geometry in which that is true, known as non-Euclidean geometry. I've talked about this a bit before, in the rerun commentary for #882.

Sometimes an axiom is so obvious that mathematicians might overlook it completely when listing their assumptions. One historical example is the axiom of choice. The axiom of choice says:

If you have a set of sets, each of those sets containing at least one object, then it is possible to select one object from every set.

Putting it less formally:

If you have a bunch of jars, each jar containing at least one marble, then it is possible to select one marble from every jar.

Obvious, right? I mean, obviously true, right? If this were not true, then something would be seriously wrong and you might suspect some sort of fatal glitch in The Matrix. But hold on. Let's put it a bit more mathematically:

If you have a set of sets, each of those sets containing at least one object, then it is possible to select one object from every set. (Noting that the sets may contain an infinite number of objects and, importantly, that the set of sets may contain an infinite number of sets.)

So imagine the first set is the set of real numbers between 0 and 1, the second set is the set of real numbers between 1 and 2, the third set is the set of real numbers between 2 and 3, and so on. Each set is infinite, and indeed there are an infinite number of sets in our collection. But it's still trivial to select one object from each set: from the first set choose 0.5, from the second choose 1.5, from the third choose 2.5, ... , from the Nth choose (N-1)+0.5, ... for each positive integer N.

This is still obviously true. It's hard to imagine a situation where the axiom of choice could possibly be false. Why would you ever not be able to choose an item from each set in a bunch of sets?

But interestingly, there is no proof of the axiom of choice. You can't prove, from other rules of mathematics, that given a collection of sets that you can choose an item from each set. But it's so obvious that you can, that many mathematicians didn't even bother making this assumption explicit, and just went on and proved a bunch of stuff, assuming you can do this.

But hold on a minute! It turns out that you can also assume the opposite of the axiom of choice - that you can't always choose an object from each set in a collection of sets - and then derive a whole other self-consistent field of mathematics based on that. It's kind of the non-Euclidean version of set theory, to borrow a term from geometry.

And in fact, if you assume the axiom of choice, then you can prove some things that sound ridiculously absurd - notably the Banach-Tarski theorem. A consequence of being able to choose one item from each set of a collection of sets is that you can cut a sphere apart and reassemble it into two spheres of the same size. A thing that is obviously true leads logically and inevitably to something that is obviously absurd. For this reason, some mathematicians now reject the axiom of choice, satisfied that it is in fact false.

In a greater sense, the axiom of choice is neither true nor false, in the same way that Euclid's parallel lines axiom is neither true nor false. You can assume it either way, and do a bunch of interesting and non-trivial mathematics either way.

[This comic was inspired by a rambling conversation between me and some friends, which actually began talking about mathematics, and for reasons unclear to me now diverged to religion and abortion politics before neatly returning full circle. I turned it on its head a bit to make the joke here.]

EDIT: Forum poster AlienAtSystem mentioned a cool way to think of the axiom of choice by proposing a game between two players:

As there are infinitely many sets, instead of going through them manually, you have to find an algorithm [to select an item from each set]. So Person A tells the other player an algorithm to select an element. Then that other player finds a set where the algorithm fails.
If you, for example say: "Take the smallest element", then I say "One of my sets is all negative integers, it doesn't have a smallest element."
Then maybe you say: "Take the smallest element, or if it does not have that, take the largest." Then I say "One of my sets is all integers, which has neither". Then maybe you add: "If it has neither, take the element closest to 0 (in case of tie, take the larger one)". Then I say: "What about the set of all integers and all their reciprocals? It doesn't have an element closest to 0 either."
When Person A gives up and says: "Look, there's some algorithm that does it, right, why do I have to recite it manually?", then that person has taken the Axiom of Choice. If they say "Okay, fine, you win, there's no algorithm you can't find a counterexample to", then that person rejects the Axiom of Choice.