The axiom of choice has a long and interesting history. Mathematics up to the early 1900s was built on (what is now called) naïve set theory: Mathematicians did not have explicit axioms for working with sets, and simply used them intuitively, with a set simply being a “collection of objects which satisfy some property.” This changed in 1901, when Bertrand Russel presented his now famous paradox, which showed that these intuitive rules for manipulating sets could cause irreconcilable mathematical contradictions. This meant a formal axiomatic system defining sets and their properties was necessary. 

The collective work of Ernst Zermelo, Abraham Fraenkel, Thoralf Skolem, and John von Neumann, spanning 1908 to 1925, led to the axiomatic set theory underpinning almost all of modern mathematics. It is called ZFC, where the final C stands for the axiom of choice (AC). The axioms of ZFC excluding AC are denoted ZF.

The controversy of AC came from its non-constructiveness – it can often be used to prove the existence of some object, without providing any way to describe that object (such as through a formula or algorithm). The situation became more complicated when the (separate) works of Kurt Gödel and Paul Cohen together showed that AC is independent of ZF. This means using the axioms of ZF, it is impossible to prove AC, but it is also impossible to disprove it. Using AC means it truly must be taken as an additional axiom.

This is not the only statement known to be independent of ZF. There are many others, such as the continuum hypothesis. However, AC is special for two reasons. The first is that it affects many areas of mathematics. For most other axioms independent of ZF, the choice to accept or negate them is usually only of immediate consequence to those working directly with said axioms. However, choosing elements from collections of sets is ubiquitous in mathematics. So, AC comes up in nearly every field, and it can easily go unnoticed  – many innocuous statements in a proof may invoke AC without it being immediately obvious why.

The second of AC’s unique traits is how AC may seem very natural or very unnatural, depending on the context. There are several statements equivalent to AC, some of which seem like they should be true in ZF, and so the fact that they must be added as an axiom is surprising. This also means that rejecting AC means rejecting these statements, which may be undesirable. At the same time, accepting AC as true can result in proving the existence of bizarre and unintuitive objects such as non-measurable sets (which are used to conclude the Banach-Tarski paradox).

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