The axiom of choice - History

Last two blogs, I introduced the axiom of choice. I will now try to present some history behind the axiom.

The story begins in 1870, when Cantor first invented transfinite numbers and the rest of set theory. This set theory makes use of the concept of well-ordered set. A set is called well-ordered if it is equiped with a natural order < such that every subset has a minimum. The standard example of a well-ordered set are the natural numbers. It is intuitively obvious that there is a well-order on there. For example {2,4,6,7} has 2 as it's minimum. A set which is not well-ordered is the set of real numbers, R. For example, the open interval ]0,3] has no smallest element. Thus we don't have a well-order.

A crucial result in set theory is the "well-ordering theorem", which state that every set can be well-ordered. That doesn't mean that every order is a well-order (indeed R with the USUAL order is not a well-order), but merely that there exists a well-order on the set. So R does carry a well-order according to the well-ordering theorem. So everything looks good, but there is just one little problem: Cantor never proved the well-ordering theorem. Indeed, he found the well-ordering theorem so obvious that he didn't see the need for a proof. Other mathematicians disagreed that it was obvious, indeed how would one construct a well-order on R? It seems impossible. So many mathematicians set out to prove the well-ordering theorem.

It was not until 1904 when a brilliant mathematican, with the name of Ernst Zermelo, proved the well-ordering theorem. However, he needed a new axiom for it: the axiom of choice. But the axiom was immediately controversial. It seems like the entire mathematical world was split in two: the opponents and the proponents of the axiom of choice. The axiom of choice became even more important when the axiom of choice was shown to be equivalent to the well-ordering theorem. And even later, Kuratowski proved the so called "Zorn's lemma" equivalent to the axiom of choice. Now, Zorn's lemma popped up everywhere in mathematics: geometry, topology, algebra, analysis,... So the axiom of choice became extremely important in mathematics, but still, nobody had given a proof for it, or even showed that it could be valid!

This situation was solved by one of the greatest minds of mathematics: Kurt Godel. Godel constructed a universe (the so-called constructible universe), where the axiom of choice holds true. Thus Godel proved that the axiom of choice is consistent with mathematics, that is, assuming the axiom of choice would not lead to any contradiction (unlike axioms like 1+1=3 or "the set of natural numbers is finite"). This means that mathematicians were now allowed to use the axiom of choice anywhere they liked. But the situation was not resolved, because perhaps the axiom of choice could be proven?

It could not, as Cohen showed with a revolutionary technique called "forcing". With forcing he constructed a universe where the axiom of choice was false. For this construction, he won the Fields-medal, the most honorable price in mathematics! Thus the axiom of choice is actually independent of mathematics. Mathematicians could choose whether they accept choice or rejected it, it's all OK from a mathematical point-of-view.

With the work of Cohen and Godel, the question of whether we should accept the axiom of choice or not, became a philosophical question. On the one hand, there are some very useful results which follow from the axiom, but on the other hand, accepting the axiom could lead to strange, disastrous things. For the following blog, we will research the various relationships that the axiom of choice has with many parts of mathematics. We will show some of the useful results which come from accepting the axiom. But we will also show the monstrous results that also follow, in particular, the infamous Banach-Tarski paradox will be discussed.

I'll see you next time!