cantor
    n 1: the musical director of a choir [syn: choirmaster,
         precentor, cantor]
    2: the official of a synagogue who conducts the liturgical part
       of the service and sings or chants the prayers intended to be
       performed as solos [syn: cantor, hazan]

Cantor \Can"tor\, n. [L., a singer, fr. caner to sing.]
   A singer; esp. the leader of a church choir; a precentor.
   [1913 Webster]

         The cantor of the church intones the Te Deum. --Milman.
   [1913 Webster]

Cantor

   1.  A mathematician.

   Cantor devised the diagonal proof of the uncountability of the
   real numbers:

   Given a function, f, from the natural numbers to the real
   numbers, consider the real number r whose binary expansion is
   given as follows: for each natural number i, r's i-th digit is
   the complement of the i-th digit of f(i).

   Thus, since r and f(i) differ in their i-th digits, r differs
   from any value taken by f.  Therefore, f is not surjective
   (there are values of its result type which it cannot return).

   Consequently, no function from the natural numbers to the
   reals is surjective.  A further theorem dependent on the
   axiom of choice turns this result into the statement that
   the reals are uncountable.

   This is just a special case of a diagonal proof that a
   function from a set to its power set cannot be surjective:

   Let f be a function from a set S to its power set, P(S) and
   let U =  x in S: x not in f(x) .  Now, observe that any x in
   U is not in f(x), so U != f(x); and any x not in U is in f(x),
   so U != f(x): whence U is not in  f(x) : x in S .  But U is
   in P(S).  Therefore, no function from a set to its power-set
   can be surjective.

   2.  An object-oriented language with fine-grained
   concurrency.

   [Athas, Caltech 1987.  "Multicomputers: Message Passing
   Concurrent Computers", W. Athas et al, Computer 21(8):9-24
   (Aug 1988)].

   (1997-03-14)