axiom of choice

Axiom of Choice

   <mathematics> (AC, or "Choice") An {axiom} of {set theory}:

   If X is a set of sets, and S is the union of all the elements
   of X, then there exists a function f:X -> S such that for all
   non-empty x in X, f(x) is an element of x.

   In other words, we can always choose an element from each set
   in a set of sets, simultaneously.

   Function f is a "choice function" for X - for each x in X, it
   chooses an element of x.

   Most people's reaction to AC is: "But of course that's true!
   From each set, just take the element that's biggest,
   stupidest, closest to the North Pole, or whatever".  Indeed,
   for any {finite} set of sets, we can simply consider each set
   in turn and pick an arbitrary element in some such way.  We
   can also construct a choice function for most simple {infinite
   sets} of sets if they are generated in some regular way.
   However, there are some infinite sets for which the
   construction or specification of such a choice function would
   never end because we would have to consider an infinite number
   of separate cases.

   For example, if we express the {real number} line R as the
   union of many "copies" of the {rational numbers}, Q, namely Q,
   Q+a, Q+b, and infinitely (in fact uncountably) many more,
   where a, b, etc. are {irrational numbers} no two of which
   differ by a rational, and

     Q+a == {q+a : q in Q}

   we cannot pick an element of each of these "copies" without
   AC.

   An example of the use of AC is the theorem which states that
   the {countable} union of countable sets is countable.  I.e. if
   X is countable and every element of X is countable (including
   the possibility that they're finite), then the sumset of X is
   countable.  AC is required for this to be true in general.

   Even if one accepts the axiom, it doesn't tell you how to
   construct a choice function, only that one exists.  Most
   mathematicians are quite happy to use AC if they need it, but
   those who are careful will, at least, draw attention to the
   fact that they have used it.  There is something a little odd
   about Choice, and it has some alarming consequences, so
   results which actually "need" it are somehow a bit suspicious,
   e.g. the {Banach-Tarski paradox}.  On the other side, consider
   {Russell's Attic}.

   AC is not a {theorem} of {Zermelo Fränkel set theory} (ZF).
   Gödel and Paul Cohen proved that AC is independent of ZF,
   i.e. if ZF is consistent, then so are ZFC (ZF with AC) and
   ZF(~C) (ZF with the negation of AC).  This means that we
   cannot use ZF to prove or disprove AC.

   (2003-07-11)