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cantor

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cantor

 

  3  definitions  found 
 
  From  Webster's  Revised  Unabridged  Dictionary  (1913)  [web1913]: 
 
  Cantor  \Can"tor\,  n.  [L.,  a  singer,  fr  caner  to  sing.] 
  A  singer;  esp.  the  leader  of  a  church  choir;  a  precentor. 
 
  The  cantor  of  the  church  intones  the  Te  Deum.  --Milman. 
 
  From  WordNet  r  1.6  [wn]: 
 
  cantor 
  n  1:  the  musical  director  of  a  choir  [syn:  {choirmaster},  {precentor}] 
  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:  {hazan}] 
 
  From  The  Free  On-line  Dictionary  of  Computing  (13  Mar  01)  [foldoc]: 
 
  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)




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