TITLE: Is the axiom of choice really all that important?
QUESTION [56 upvotes]: According to this book:

The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern algebra, analysis, mathematical logic, and topology (often under the name Zorn's Lemma).

I am not a set theorist, and I don't pretend to be, but I have heard of some weird things that can happen with choice, such as the Banach–Tarski paradox--paradoxes like these are presumably why the Axiom of Choice was so controversial at first, but I am interested in what would happen without choice.

Question: What notable consequences would occur without the Axiom of Choice?

I found one very interesting example here on an MO thread (reproduced here for ease):

The universe can be very a strange place without choice. One consequence of the Axiom of Choice is that when you partition a set into disjoint nonempty parts, then the number of parts does not exceed the number of elements of the set being partitioned. This can fail without the Axiom of Choice. In fact, if all sets of reals are Lebesgue measurable, then it is possible to partition $2^{\omega}$ into more than $2^{\omega}$ many pairwise disjoint nonempty sets!

What other weird things would result without this axiom? Would it really be a devastating blow to mathematics or is it really not that big of a deal? I'm hoping for examples/answers geared toward the undergrad level--suitable for someone with very little set theory background.

REPLY [8 votes]: The axiom of choice is needed to construct a function $f:[0,1]\rightarrow R$, such that $f(r^n)=f(r)$ for $r\in[0,1]$ and all natural numbers $n$, that is a counterexample for the uniqueness in a vibrating string inverse problem.