Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some important consequences of a strong formulation of the axiom of choice?

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Each of the following is equivalent to the Axiom of Choice:

  • Every vector space (over any field) has a basis.

  • Every surjection has a right inverse.

  • Zorn’s Lemma.

The first is extremely important and useful. The third is used all the time, especially in algebra, also very important and useful.

You could write an entire book on important consequences (and equivalents) of the Axiom of Choice.

Unfortunately, any publisher worth his salt would reject it, since both have already been written:

  1. Rubin, Herman; Rubin, Jean E.
    Equivalents of the axiom of choice. North-Holland Publishing Co., Amsterdam 1963 xxiii+134 pp.

  2. Rubin, Herman; Rubin, Jean E.
    Equivalents of the axiom of choice. II.
    Studies in Logic and the Foundations of Mathematics, 116. North-Holland Publishing Co., Amsterdam, 1985. xxviii+322 pp. ISBN: 0-444-87708-8

  3. Howard, Paul; Rubin, Jean E.
    Consequences of the axiom of choice. Mathematical Surveys and Monographs, 59. American Mathematical Society, Providence, RI, 1998. viii+432 pp. ISBN: 0-8218-0977-6

These books are probably not the best place to start, though; the first book is okay, listing some of the most important equivalents as they were known before Cohen’s work, but at least the last is pretty difficult to slough through.

If you want a good introduction to the Axiom of Choice and some idea of its uses, Horst Herrlich’s Axiom of Choice, Lecture Notes in Mathematics v. 1876, Springer-Verlag (2006) ISBN: 3-540-30989-6 is pretty good, discussing some of the bad things that happen if you don’t have AC, some of the bad things that happen if you do have AC, and some alternative axioms that contradict AC but lead to very nice theorems.

Axiom of Choice $\iff$ A non-empty product of non-empty sets is non-empty.

The Axiom of choice is required to prove for example that every ring has a maximal ideal (or a minimal prime), or the existence of an algebraic closure.

There Lebesgue measure is non-trivial – there are non-measurable sets in $\mathbb{R}$.

The Hahn-Banach theorem and Tychonoff’s theorem are two major ways in which the axiom of choice gets used in analysis (I am echoing other answers here because these are really important). In addition to Zorn’s lemma, the third “classic” theorem known to be equivalent to AC is the well-ordering theorem.

Here’s a somewhat surprising one I was recently made aware of: you cannot prove that a countable union of countable sets is countable without some form of countable choice. The problem is that in order for any of the standard proofs to go through, you need to choose, for each countable set in the union, a bijection of it with $\mathbb{N}$, and there is no way to do this canonically in general. (Fortunately, for most countable sets we encounter in practice we can define explicit bijections, so there’s usually no problem.)

A weak form of choice, the Boolean prime ideal theorem (or equivalently the ultrafilter lemma), also has many useful consequences. For example, it is equivalent (I think) to Tychonoff’s theorem for compact Hausdorff spaces. It is equivalent to Gödel’s completeness theorem, as well as to the compactness theorem. It allows you to construct ultraproducts, which has a whole genre of uses such as, for example, defining nonstandard models of $\mathbb{R}$ or more exotic things. Terence Tao has written several good blog posts, including this one, on these and related subjects.

You might be interested in browsing http://consequences.emich.edu/CONSEQ.HTM for more consequences of AC. There are really quite a lot.

I think that one of the most important implications of the axiom of choice is actually the equivalence of continuity between the Cauchy definition of $\epsilon$-$\delta$ and the Heine definition using sequences.

A consequence that motivates me is: If $I$ is some non-empty class and $X_i$ is a non-empty class $i\in I$, then $$\prod_{i\in I}X_i\ne\emptyset$$

EDIT

An other consequence that motivates me is: If $I$ is some non-empty set and $X_i$ is a compact topological space $i\in I$, then $\prod_{i\in I}X_i\ne\emptyset$ is compact.

Of course these are known equivalences of the Axiom of choice.

To construct left- and right inverses of functions where the domain is not well-ordered we need the axiom of choice. Also, we can always get a well-ordering (this is equivalent to the axiom of choice) so that way we also could obtain this result (then you take for the many inverses the smallest one).

Also, it is important in functional analysis, for example Hahn-Banach depends on it. The axiom of choice implies Hahn-Banach but not the other way around.

Adding a 2nd answer in order to keep things to one example per answer.

There was some time ago a question on MO about whether Arzelà-Ascoli requires choice, and the answer turns out to be that it indeed does require choice – though not the full strength of AC.