### General

The Axiom of Choice states that given any collection of "bins", it is possible to choose an object from each one of the bins. This is the more intuitive version of the axiom. Other versions include the well-ordering principle, or Zorn's lemma. In fact, many useful results of mathematics are derived from (and often equivalent to) the axiom of choice, which makes it a hard axiom to ignore in any field where mathematics is used. The comic's caption illustrates used of the axiom of choice very well: the well-ordering principle states the existence of a certain kind of ordering on any set, without giving any detail as to what this ordering looks like.

The Axiom of Choice, which is referenced in this comic, implies the Banach-Tarski Theorem (paradox), which states that it is possible to cut a sphere in a finite number of pieces, and reassemble the pieces into two spheres, each one of volume equal to the first one. It is called a paradox because it is counter-intuitive.