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 nonempty 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 BanachTarski 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.
(20030711)
Nearby terms:
Axiomatic Architecture Description Language «
axiomatic semantics « axiomatic set theory «
Axiom of Choice » Axiom of Comprehension » AXLE
» ayacc
