catflap.org Online Dictionary Query

  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
          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 Frankel set theory (ZF).
          Godel 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)
       
       

  axiom of choice
     n.
     (lb en set theory) One of the axioms of set theory, equivalent to the
  statement that an arbitrary direct product of non-empty sets is
  non-empty; any version of said axiom, for example specifying the
  cardinality of the number of sets from which choices are made.

  axiom of choice
     n.
     (lb en set theory) One of the axioms of set theory, equivalent to the
  statement that an arbitrary direct product of non-empty sets is
  non-empty; any version of said axiom, for example specifying the
  cardinality of the number of sets from which choices are made.

  axiom of choice
     n.
     (lb en set theory) One of the axioms of set theory, equivalent to the
  statement that an arbitrary direct product of non-empty sets is
  non-empty; any version of said axiom, for example specifying the
  cardinality of the number of sets from which choices are made.

  axiom of choice
     n.
     (lb en set theory) One of the axioms of set theory, equivalent to the
  statement that an arbitrary direct product of non-empty sets is
  non-empty; any version of said axiom, for example specifying the
  cardinality of the number of sets from which choices are made.

  axiom of choice
     Engelska n.
     (tagg språk=en matematik) urvalsaxiomet

  axiom of choice /ˈaksɪəm ɒv tʃˈɔɪs/
  Auswahlaxiom  [math.]

  axiom of choice /ˈaksɪəm ɒv tʃˈɔɪs/ 
  valinta-aksiooma
  axiom that any product of non-empty sets is non-empty

  axiom of choice /ˈaksɪəm ɒv tʃˈɔɪs/
  aksiom biranja, aksiom izbora

  axiom of choice /ˈaksɪəm ɒv tʃˈɔɪs/ 
  選出公理, 選択公理
  axiom that any product of non-empty sets is non-empty

  axiom of choice /ˈaksɪəm ɒv tʃˈɔɪs/ 
  urvalsaxiom
  axiom that any product of non-empty sets is non-empty