TITLE: Can you explain the "Axiom of choice" in simple terms?
QUESTION [139 upvotes]: As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. 

I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this, the Wikipedia entry is not in plain, simple, understandable language. Can someone give me a nice simple explanation of what this axiom is, and perhaps explain the XKCD joke as well?

REPLY [12 votes]: You have some boxes, at least one. There may be only one box, seven boxes, or infinitely many boxes.
In each and every box, there are some items, at least one. There may be arbitrarily many items.

Question: Can you pick, at least in principle, exactly one item from each box, without necessarily describing the details or rules of how you select them?
The axiom of choice says that the answer to this question is affirmative.

It might seem quite innocuous, but it turns out that the axiom of choice has groundbreaking consequences in virtually all subfields of mathematics. Some of these consequences are fundamental results, others are mind-boggling paradoxes. The long list includes:

[Geometry] Banach–Tarski paradox. (The axiom of choice makes it possible to cut an object into a finite number of pieces in such a weird way that you can reassemble two copies of the same object of the same size!)
[Topology] Tychonoff’s theorem.
[Algebra] Every non-trivial vector space has a Hamel basis.
[Order theory] Zorn’s lemma.
[Order theory] Well-ordering theorem and transfinite induction.
[Set theory] Countable union of countable sets is countable.
[Set theory] Every infinite set has a proper subset with the same cardinality.
[Measure theory] Existence of sets that are not Lebesgue measurable.
[Functional analysis] Baire’s category theorem.
[Functional analysis] Hahn–Banach theorem.
[Functional analysis] Every Banach space admits linear maps that are nowhere continuous.
[Functional analysis] Krein–Milman theorem.