The DICT Development Group

Search for:
Search type:

Database copyright information
Server information

1 definition found
 for Axiom of Choice
From The Free On-line Dictionary of Computing (30 December 2018) :

  Axiom of Choice
      (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
     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.

Contact=webmaster@dict.org Specification=RFC 2229