Axiom Of Choice

   In [1]mathematics (specifically [2]set theory) axiom of choice is a
   possible [3]axiom which basically states we can arbitrarily choose
   elements of sets and which is famous for being controversial and
   problematic because it causes trouble both when we accept or reject it.
   Now it's actually been included in [4]ZFC, a kind of "commonly used base
   for mathematics", but its controversial nature stands. Note that this
   topic can go to a great depth and lead to philosophical debates, there is
   a huge rabbit hole and mathematicians can talk about this for hours; here
   we'll only state the very basic and quite simplified things, mostly for
   those who aren't professional mathematicians but need some overview of
   mathematics (e.g. programmers).

   Indeed, what really IS the axiom of choice? It is an [5]axiom, i.e.
   something that we can't prove but can either accept or reject as a basic
   fact so that we can use it to prove things. Informally it says that given
   any collection of sets (even an infinite collection of infinitely large
   sets), we can make an arbitrary selection of one element from each set.
   More mathematically it says: if we have a collection of sets, there always
   exists a [6]function f such that for any set S from the collection f(S) is
   an element of S.

   This doesn't sound weird, does it? Well, in many normal situations it
   isn't. For example if we have finitely many sets, we can simply write out
   each element of the set, we don't need to define any selection function,
   so we don't need axiom of choice to make our choice of elements here. But
   also if we have infinitely many sets that are well ordered (we can compare
   elements), for example infinitely many sets of [7]natural numbers, we can
   simply define a function that takes e.g. the smallest number from each set
   -- here we don't need axiom of choice either. The issues start if we have
   e.g. infinitely many sets of [8]real numbers (which can't be well ordered
   without the axiom of choice, consider that e.g. open intervals don't have
   lowest number) -- here we can't say how a function should select one
   element from each set, so we have to either accept axiom of choice (we say
   it simply can be done "somehow", e.g. by writing each element out on an
   infinitely large paper) or reject it (we say it can't be done). Here it is
   again the case that what's normally completely non-problematic starts to
   get very weird once you involve [9]infinity.

   Why is it problematic? Once you learn about axiom of choice, your first
   question will probably be why should it pose any problems if it just seems
   like an obvious fact. Well, it turns out it leads to strange things. If we
   accept axiom of choice, then some weird things happen, most famously e.g.
   the [10]Banach-Tarski paradox which uses the axiom of choice to prove that
   you can disassemble a sphere into finitely many pieces, then move and
   rotate them so that they create TWO new spheres, each one identical to the
   original (i.e. you duplicate the original sphere). But if we reject the
   axiom of choice, other weird things happen, for example we can't prove
   that every vector space has a basis -- it seems quite elementary that
   every vector space should have a basis, but this can't be proven without
   the axiom of choice and in fact accepting this implies the axiom of choice
   is true. Besides this great many number of proofs simply don't work
   without axiom of choice. So essentially either way things get weird,
   whether we accept axiom of choice or not.

   So what do mathematicians do? How do they deal with this and why don't
   they kill themselves? Well, in reality most of them are pretty chill and
   don't really care, they try avoid it if they can (their proof is kind of
   stronger if it relies on fewer axioms) but they accept it if they really
   need it for a specific proof. Many elementary things in mathematics
   actually rely on axiom of choice, so there's no fuss when someone uses it,
   it's very normal. Turns out axiom of choice is more of something they
   argue over a beer, they usually disagree about whether it is INTUITIVELY
   true or false, but that doesn't really affect their work.

Links:
1. math.md
2. set_theory.md
3. axiom.md
4. zfc.md
5. axiom.md
6. function.md
7. natural_number.md
8. real_number.md
9. infinity.md
10. banach_tarski.md